The following table lists metric prefixes that come up frequently in physics. Learning these prefixes will help you in the various exercises.


 Magdalene Isabella Wilkins
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1 Chapter 0 Solutions Circles Learning Goal: To calculate the circumference or area of a circle Every day, we see circles in compact disks, coins, and wheels, just to name a few examples Circles are also very common in math and physics, because they are easy to describe and calculate with There is a very simple definition of a circle: Pick a point A and a distance r A circle with center A and radius r is the set of all points a distance r from the point A For instance, if you pick the center of your screen as A and 3 cm s r, then all of the points on the screen that are 3 cm away from the center form a circle An easy way to draw a circle is to hold a string fixed at one end (for instance, with a pin) and attach a pen to the other end Holding the string taut and drawing wherever the taut string allows gives a circle This is the same technique you follow when using a compass With a compass, the string is simply replaced by a metal or plastic structure, which usually has some markings to let you pick the radius of your circle The radius of a circle is the only measure that you need to determine any other measure, such as the circumference or area For instance, the circumference C of a circle (the length measured around the outside of the circle, ie, the perimeter) is C=2πr Suppose that you have a piece of string 7 cm long If one end is held fixed and you draw with a pen at the other end, keeping the string tight, then you will draw a circle Part A What is the circumference of this circle? Express your answer in centimeters to three significant figures C= 440 cm
2 Suppose that you needed to make a pen for some small animals You have 12 m of fencing You decide to make a circular pen, because if you wish to enclose an area using a given length of fencing, then a circular fence encloses a larger area than a fence of any other shape Part B What is the radius r of the pen? Express your answer in meters to four significant figures r= 1910 m Part C What is the area A of the circular enclosure from Part B? Express your answer in square meters to three significant figures A= 115 m 2 In Part B, it was stated that a circular enclosure gives the greatest area for a given perimeter If you wrote the areas of an equilateral triangle, a square, and a circle for a given perimeter P, you would find, and, The formula for the circle has the smallest denominator and thus the largest area for the given perimeter This is true regardless of what shapes you choose to compare to the circle Part D If you swim from one point at the edge of the pool to another, along a straight line, what is the longest distance d you can swim? Express your answer in meters to three significant figures Hint 1 How to approach the problem The longest distance between two points on a circle is the diameter of the circle The diameter is a line through the center of the circle Think about how the length of such a line relates to the radius Drawing a picture may help if you are having trouble seeing how the two relate You have the area of the pool and would like to know the radius Since the formula A=πr 2 relates the area to the radius, you can use this formula to find the radius, using your given value of the area
3 Hint 2 A helpful picture The picture below shows a diameter of a circle, which is the longest segment that can be drawn connecting two points on a circle Notice that since the diameter goes through the center of the circle, you can break it up into two radii d= 564 m
4 Converting Units The ability to convert from one system of units to another is important in physics It is often impractical to measure quantities in the standard meters, kilograms, and seconds, but the laws of physics that you learn will involve constants that are defined in these units Therefore, you may often have to convert your measured quantities into meters, kilograms, and seconds The following table lists metric prefixes that come up frequently in physics Learning these prefixes will help you in the various exercises mega ( ) kilo ( ) centi ( ) milli ( ) micro ( ) nano ( ) When doing unit conversions, you need a relation between the two units For instance, in converting from millimeters to meters, you need to know that Once you know this, you need to divide one side by the other to obtain a ratio of m to mm If you are converting from millimeters to meters, then this is the proper ratio It has mm in the denominator, so that it will cancel the units of the quantity that you are converting For instance, if you were converting 63 mm then you would have If you were converting a quantity from meters to millimeters, you would use the reciprocal ratio:
5 Part A Suppose that you measure a pen to be 105 cm long Convert this to meters Express your answer in meters Hint 1 Relating centimeters and meters To solve this problem, you will need to use the relation 100 cm = 1 m You can determine such relations using the metric prefixes given in the introduction to this problem If one centimeter equals 102 meters, then you need 10 2 centimeters to equal a whole meter, just as you know that if one quarter equals 41 US dollars, then you need 4 1 quarters to equal a whole US dollar 105 cm = 0105 m Part B Suppose that, from measurements in a microscope, you determine that a certain bacterium covers an area of 150 µm 2 Convert this to square meters Express your answer in square meters Hint 1 Find the conversion factor Which of the following gives the proper conversion factor to use? From the table in the introduction, you can see that 1 µm = 106 m, which gives 10 6 µm = 1 m 150 µm 2 = m 2 Part C Suppose that you find the volume of all the oceans to be 14 X 10 9 m 3 in a reference book To find the mass, you can use the density of water, also found in this reference book, but first you must convert the volume to cubic meters What is this volume in cubic meters? Express your answer in cubic meters Hint 1 Find the conversion factor Which of the following gives the proper conversion factor to use? From the table in the introduction, you can see that 1 km = 10 3 m 14 x10 9 km 3 = m 3 Part D In a laboratory, you determine that the density of a certain solid is 523x106 kg/mm 3 Convert this density into kilograms per cubic meter Notice that the units you are trying to eliminate are now in the denominator The same principle from the previous parts applies: Pick the conversion factor so that the units cancel The only change is that now the units you wish to cancel must appear in the numerator of the conversion factor Express your answer in kilograms per cubic meter Hint 1 Find the conversion factor Which of the following gives the proper conversion factor to use? From the table in the introduction, you can see that 1 mm = 103 m Recall that you are trying to cancel units out of the denominator of the fraction
6 = 5230 You are now ready to do any sort of unit conversion You may encounter problems that look far more complex than those you've done in this problem, but if you carefully set up conversion factors one at a time to cancel the units you don't want and replace them with the units that you do want, then you will have no trouble
7 Dealing with Numbers from Tiny to Huge Learning Goal: To understand powers of 10 and scientific notation Physics deals with an incredible range of objects and phenomena: from processes involving subatomic particles to those involving the entire universe This variety leads to sizes from the width of an atom to the distances between stars Physicists use several methods to make dealing with very large or very small numbers more convenient We will start exploring these methods by considering two numbers: the lifetime of a particle called a top quark and the age of the universe The age of the universe is around 100,000,000,000,000,000 s A top quark has a lifetime of roughly s Writing numbers out with all these zeros is not very convenient Such quantities are usually written as powers of 10 The age of the universe can be written as s and the lifetime of a top quark as s Note that means the number you would get by multiplying 10 times 10 times 10 a total of 17 times This number, as you can see above, would be a one followed by seventeen zeros Similarly, is the result of multiplying 01 (or 1/10) times itself 24 times As seen above, this is written as 23 zeros after the decimal point followed by a one Part A How many top quark lifetimes have there been in the history of the universe (ie, what is the age of the universe divided by the lifetime of a top quark)? Note that these powers of 10 follow the same rules that any exponents would follow Express your answer numerically Hint 1 Multiplying and dividing with exponents Recall that when you multiply two numbers with the same base (eg, 2 4 and 2 7 ), you obtain the result by adding the exponents (eg, 2 4 * 2 7 = = 2 11 ) When you divide, you subtract the exponents For example, Part B Compute (429x10 15 )*(151x104 ) Express your answer to three significant figures
8 If you had been asked to multiply 429x10 15 times 300x104, you would have found (to three significant figures) 129x10 11 You should rewrite this in the more accepted form 129x10 12 (Notice that you must increase the exponent to balance moving the decimal point to the left) Part C Compute 628x x10 11 Express your answer to three significant figures Hint 1 How to approach the problem Of the two powers of 10, 11 is the smallest Convert 830x10 11 to some number times 10 13, then add as usual Part D Suppose that a certain fortunate person has a net worth of $760 billion ($760x10 10 ) If his stock has a good year and gains $320 billion (320x10 9 ) in value, what is his new net worth? Express your answer to three significant figures Hint 1 How to approach the problem Of the two powers of 10, 9 is the smallest Convert 320x10 9 to some number times 10 10, then add as usual dollars Part E Suppose that this individual now decides to give oneeighth of a percent of his new net worth to charity How many dollars are given to charity? Express your answer to three significant figures Hint 1 Percent and scientific notation Recall that 50 percent is the same as 50/100 or 50 x 102 Knowing that you can rewrite "oneeighth of a percent" as 0125x102 = 125x dollars
9 Interpreting Graphs Learning Goal: To be able to gain many different types of information from a graph Being able to read graphs is an important skill in physics It is also critical in daytoday life, as information in the news and in business meetings is often presented in graphical form In this problem, you will consider a single graph and all of the information that can be gained from it Since the graph axes have no labels, think of it as a graph of something important to you, whether that is GPA, your bank balance, or something else Specific applications will be noted for each way of analyzing a graph The easiest information to obtain from a graph is its value at a point The height of the graph above the horizontal axis gives the value of the graph Points above the horizontal axis have positive values, whereas points below the axis have negative values The vertical axis will usually have specific values marked off so that you can tell exactly what value each height corresponds to In the graph you've been given, there are no exact values labeled, but you can still tell relative values; you can make statements such as, "At point D, the graph has a greater value than at point C" Part A At which point(s) does the graph have a positive value? Enter all of the correct letters in alphabetical order For instance, if you think that the correct choices are B and F, you would enter BF Hint 1 Determining positive values from a graph Any value above the horizontal axis is positive; any below is negative Where the graph intersects the horizontal axis, the value of the graph is zero, which is neither positive nor negative ABCDEF The difference between positive and negative is important in many situations, for instance on your bank statement In physics it makes a big difference in many scenarios Positive position means to the right or above some reference point; negative position means to the left or below the reference point Positive velocity means moving to the right, whereas negative velocity means moving to the left
10 The graph is often more convenient than a table of numbers or an equation, because you can immediately see where the graph takes positive values and where it takes negative values With an equation or a table of numbers, this would take some algebra or guess work Part B At which point does the graph have its maximum value? Enter the correct letter E Part C Look at the graph from the introduction The three points C, D, and F are all on straight segments Rank them from greatest rate of change to least rate of change Hint 1 Slope at point D When the graph is horizontal, the slope of the graph is zero You can see this by noticing that the change in vertical position as you move from left to right across a horizontal graph is zero No matter what the change in horizontal position is, when you divide to find the slope it will be zero Part D At which point is the graph increasing at the greatest rate? For now, ignore point E We will discuss it after this part Enter the correct letter Hint 1 Drawing the tangent To find the tangent line at a particular point, you should draw a dot at that point on the graph and then draw pairs of points, one on either side of the point you care about, that are the same distance from the point you care about If you then connect those pairs of dots, the lines connecting them will get closer and closer to the proper slope as you move to pairs that are closer and closer to the point you care about Once you get pretty close to that point, you should be pretty confident of the slope for the tangent line
11 With practice, you will gain an intuitive ability to see roughly how the slope of the tangent at a point should look D You were told to ignore point E for this part This is because the rate of change is not well defined at sharp corners You won't ever be asked for the rate of change of a graph at a sharp corner, though points near the corner should have welldefined rates of change Points B and C are also special, because the slope at those points is zero This should be easy to see at C, since the graph is actually a horizontal line in the area near C If you carefully work out the tangent at point B using the method described in the hint for this part, you will see that the tangent is horizontal there as well Since a horizontal tangent has a slope of zero, which is neither negative nor positive, the graph is neither increasing nor decreasing at points B and C Part E At which point(s) is the graph decreasing? Enter all of the correct letters in alphabetical order For instance, if you think that the correct choices are B and F, you would enter BF FGH Part F You wish to find the area under the graph between the origin and some point on the graph Which point will yield the greatest area? Enter the correct letter G Part G You are looking at the area under the graph between two points The area is zero Which two points are you looking at?
12 Enter the two letters in alphabetical order For instance, if you think that the correct choice is B and F, you would enter BF Hint 1 How to approach the problem Since you want the area under the curve between two points to be zero, the graph must define equalsized shaded regions above and below the horizontal axis Look for two points that are near each other, one above and one below the horizontal axis FH
13 Multiplying and Dividing Fractions Learning Goal: To understand the multiplication and division of fractions If you had eight quarters, you could likely figure out relatively quickly that this amounted to two dollars Although this may be purely intuitive, the underlying math involves multiplication of fractions The value of eight quarters is the same as When you multiply fractions, you multiply the numerators (the top numbers in the fractions) to get the numerator of the answer, and then multiply the denominators (the bottom numbers in the fractions) to find the denominator of the answer In this example, giving 2, as you expected Similarly, asking for a fraction of a fraction (eg, "one fifth of a quarter") is a case of multiplying fractions:, In this problem, before entering your answer, be sure to reduce your fraction completely If you get 8/ 6 for your answer, reduce it to 4/3 before entering it, or else it will be marked wrong Also, don't worry if the numerator is larger than the denominator It is almost always easier and more useful to further calculations to leave such answers as improper fractions rather than to convert them to mixed numbers such as 1&1/3 Part A If you have a quarter of a pie and you cut it in half, what fraction of a pie would each slice represent? Give the numerator followed by the denominator, separated by a comma Hint 1 Setting up the equation The problem is asking you for the value of "half of a quarter" This translates into math as Multiply the numerators to find the numerator of the answer Then, multiply the denominators to find the denominator of the answer 1, 8
14 Part B Find the value of Though these numbers aren't quite as nice as the ones from the example or the previous part, the procedure is the same, so this is really no more difficult Give the numerator followed by the denominator, separated by a comma Hint 1 Find the numerator To find the numerator, simply multiply the numerators of the two fractions (17 and 2) What is the result? 34 Hint 2 Find the denominator To find the denominator, simply multiply the denominators of the two fractions (15 and 11) What is the result? , 165 Dividing fractions is no more difficult than multiplying them Consider the problem Notice that division by 3 is identical to multiplication by 1/3, because both operations consist of breaking the first number into three parts Thus The only new step in division of fractions is that you must invert (flip) the second fraction Then, simply multiply as shown here Part C Consider the following equivalent expressions: and What are the values of a and b? Give the value of a followed by the value of b, separated by a comma
15 12, 13 Part D Calculate the value of Give the numerator followed by the denominator, separated by a comma 9, 26 Part E Now, find the value of Don't be intimidated by the complexity of this expression Finding this value consists of simply multiplying twice and then dividing once, tasks that are no more difficult than what you've done before Give the numerator followed by the denominator, separated by a comma Hint 1 How to approach the problem Break the problem down into individual operations Notice that is the same as This second form makes the individual operations more obvious First, multiply Then, multiply
16 Finally, divide the two fractions that you have found Hint 2 Find the value of 3/16 * 2/5 What is the value of 3/16 * 2/5? Give the numerator followed by the denominator, separated by a comma 3, 40 Hint 3 Find the value of 7/4 * 3/2 What is the value of 7/4 * 3/2? Give the numerator followed by the denominator, separated by a comma 21, 8 1, 35
17 Special Triangles Learning Goal: To understand the properties of special types of triangle There are several special types of triangle that you will encounter often in physics Understanding the properties of special triangles is particularly useful in working with vectors but may also arise in other contexts such as optics Part A An equilateral triangle is a triangle with all three sides of equal length All of the angles in an equilateral triangle are equal What is the measure of angle θ in the triangle shown? equals 180 Recall that the sum of the angles in a triangle Express your answer in degrees 60 Since the sum of the angles in any triangle is 180, all equilateral triangles will have three 60 angles Part B
18 Dividing an equilateral triangle in half leads to the next special triangle that we will look at Consider the figure showing an equilateral triangle with a line dividing it down the middle This line divides both the angle at the top of the triangle and the base of the triangle into two equal parts Find the measure of angle ϕ in degrees and the length of segment x in centimeters Express the two answers in degrees and centimeters, respectively Separate the two with a comma ϕ, x= 30, 7 degrees, cm Dividing an equilateral triangle in half, as we just did here, yields a triangle often called a right triangle, in reference to the measures of its acute angles in degrees Part C What is the length y of the remaining, vertical side of the right triangle? Express your answer in centimeters to three significant figures Hint 1 Use the Pythagorean Theorem You can find the value of y using the Pythagorean Theorem The Pythagorean Theorem says that a 2 + b 2 = c 2, where a and b are the lengths of the legs of a right triangle and c is the length of the
19 hypotenuse (the side opposite the right angle) Which of the following gives the proper values of a, b, and c for this triangle? = 121 In a right triangle with acute angles of measure 30 and 60, the sides will always be in the ratio 1: : 2, for the sides opposite the 30, 60, and 90 angles, respectively Part D In a right triangle, the two legs have the same length In the figure length of the hypotenuse of this triangle? both are given lengths of 1 What is the
20 Express your answer in centimeters to four significant figures Hint 1 Use the Pythagorean Theorem You can find the value of x using the Pythagorean Theorem The Pythagorean Theorem says that a 2 + b 2 = c 2, where a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse (the side opposite the right angle) Which of the following gives the proper values of a, b, and c for this triangle? 1414 In a right triangle with two acute angles of measure 45, the sides will always be in the ratio 1: 1:, for the sides opposite the 45, 45, and 90 angles, respectively
21 Pythagorean Theorem Learning Goal: To understand and apply the Pythagorean Theorem The Pythagorean Theorem is named after a religious school from ancient Greece whose students believed whole numbers to be the foundations of the universe They discovered much interesting math using whole numbers However, the discovery that they are most famous for also led to the downfall of their religion! The Pythagorean Theorem leads directly to the discovery of irrational numbers numbers that cannot be written as the ratio of two whole numbers Seeing that even something as simple as the diagonal of a square leads to irrational numbers shattered their belief in the holiness of whole numbers, but this insight also laid the foundation for many of the discoveries that made Greek mathematics, particularly geometry, so successful The Pythagorean Theorem relates the lengths of the two legs (the sides opposite the two acute angles) a and b of a right triangle to the length of the hypotenuse (the side opposite the right angle) c Given a right triangle as shown in the figure, the Pythagorean Theorem is written For instance, if you had a right triangle with legs both of length 1 (ie, a=b=1), then the Pythagorean Theorem would give so that c= Part A Now, consider a right triangle with legs of lengths 5 cm and 12 cm What is the length c of the hypotenuse of this triangle? Express your answer to three significant figures c= 13 cm,
22 Part B Suppose that you have measured a length of 6 cm on one board and 8 cm on the other You would adjust the two boards until the length of the string had value c to ensure that the boards made a right angle What is c? Express your answer in centimeters to three significant figures c= 10 cm A set of three integers that form the legs and hypotenuse of a right triangle is called a Pythagorean triple Any Pythagorean triple multiplied by an integer is another Pythagorean triple For instance, since this problem showed you that 6, 8, 10 is a Pythagorean triple, you also know that 12, 16, 20 is a Pythagorean triple (found by doubling 6, 8, and 10) Part C Use the Pythagorean Theorem to determine which of the following give the measures of the legs and hypotenuse of a right triangle Check all that apply 3, 4, 5 4, 11, 14 9, 14, 17 8, 14, 16 8, 15, 17 Part D What is the length x of a side of the small inner square? Express your answer in terms of the variables a and b Hint 1 A more helpful figure In the figure below, the side of length a for the yellow triangle plus the side of the small square makes up the side of length b for the light blue triangle You can set up a simple equation relating a, b, and x using this fact Solve this equation for x
23 x= b a Part E Given that the side of the square has a length b  a, find the area of one of the four triangles and the area of the small inner square Give the area of one of the triangles followed by the area of the small inner square separated by a comma Express your answers in terms of the variables a and b Hint 1 Finding the areas Recall that the formula for the area of a triangle is A = (1/2)bh, where b is the length of the base and h is the height For a right triangle, the two legs form the base and height Also, recall that the formula for the area of a square is A = s 2, where s is the length of a side of the square =, Since the entire square is made up of four such triangles plus the small square, the area of the entire square is Since the entire square has sides of length c, you could also simply write Equating the two expressions for the area of the entire square gives the Pythagorean Theorem!,
24 The Root Is the Problem Learning Goal: To understand and be able to perform arithmetic involving square roots Many times in physics, you will deal with quantities involving square roots It is important that you be able to easily handle such quantities, so that difficulties in arithmetic don't hinder your understanding of the physics that leads to them The standard form for a number involving a square root is a, where b is a number that cannot be divided by a perfect square The formula a means "a multiplied by the square root of b" Don't think of 3 as being any more complicated than 3*2, because both are just simple multiplication Given an expression such as, where the number under the square root sign is divisible by a perfect square, you can transform it into the standard form To do this you will need to make use of an important property of square roots: This says that you can break up the number under the square root into the perfect square and the leftovers For example (using the fact that 4= 2*2), Part A Express in the form a Give the value of a followed by the value of b separated by a comma Hint 1 Factors of 75 The prime factorization of 75 is 5, 3 Part B Find the product of and 6 Express it in standard form (ie, a ) Give the value of a followed by the value of b separated by a comma Hint 1 Multiply the two numbers before reducing Think of the two numbers as 1 and 6, so that you can see both coefficients and both radicals explicitly Find the product of the coefficients (1 and 6) and the product of the numbers under the square roots (30 and 10) Enter the product of the coefficients followed by the product of the numbers under the square roots separated by a comma
25 6, 300 Now you know that Look at the factors of 300 to determine whether it is divisible by a perfect square 60, 3 Part C Which of the following can be reduced to a single number in standard form? Part D Find in standard form Give the value of a followed by the value of b separated by a comma Hint 1 Express in standard form The prime factorization of 126 is Using this, express 126 in standard form Give the value of a 3, 14 Hint 2 Express in standard form The prime factorization of 56 is Using this, express 56 in standard form followed by the value of b separated by a comma Give the value of a followed by the value of b separated by a comma 2, 14 5, 14
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27 Proportional Reasoning Learning Goal: To understand proportional reasoning for solving and checking problems Proportional reasoning involves the ability to understand and compare ratios and to produce equivalent ratios It is is a very powerful tool in physics and can be used for solving many problems It's also an excellent way to check answers to most problems you'll encounter Proportional reasoning is something you may already do instinctively without realizing it Part A You are asked to bake muffins for a breakfast meeting Just as you are about to start making them, you get a call saying that the number of people coming to the meeting has doubled! Your original recipe called for three eggs How many eggs do you need to make twice as many muffins? Express your answer as an integer 6 Part B You have a dozen eggs at home, and you know that with them you can make 100 muffins If you find that half of the eggs have gone bad and can't be used, how many muffins can you make? Express your answer as an integer 50 Recall that dividing a variable is the same as multiplying it by a fraction If you keep this in mind, then you can change this problem from "the number of eggs are divided by two" into "the number of eggs are multiplied by onehalf," which works just as any other multiplication If you look at the graph for the linear relationship, dividing by 2 is like moving from the middle point to the left point marked on the graph Part C When sizes of pizzas are quoted in inches, the number quoted is the diameter of the pizza A restaurant advertises an 8inch "personal pizza" If this 8inch pizza is the right size for one person, how many people can be fed by a large 16inch pizza? Express your answer numerically Hint 1 How to approach the problem The area of a pizza is what determines how many people can be fed by the pizza You know that the area of a circle is proportional to the square of the radius Since the radius is proportional to the diameter, it follows that the area is also proportional to the square of the diameter Use this relation to determine how the area, and therefore the number of people fed, changes 4 Part D If a car is speeding down a road at 40 miles/hour (mph), how long is the stopping distance D 40 compared to the stopping distance D 25 if the driver were going at the posted speed limit of 25?
28 Express your answer as a multiple of the stopping distance at 25 mph Note that D 25 is already written for you, so just enter the number Hint 1 Setting up the ratio Since 40/25 = 16, the car is moving at a speed 16 times the speed limit of 25 mph The stopping distance is proportional to the square of the initial speed, so the stopping distance will increase by a factor of the square of 16 = 256 The quadratic relationship between stopping distance and initial speed is part of the reason that speeding fines are doubled in school zones: At low speeds, a small change in speed can lead to a large change in how far your car travels before it stops Part E A construction team gives an estimate of three months to repave a large stretch of a very busy road The government responds that it's too much inconvenience to have this busy road obstructed for three months, so the job must be completed in one month How does this deadline change the number of workers needed? Hint 1 The proportionality The time to complete the job should be inversely proportional to the number of workers on the job Therefore, reducing the time by a factor of 3 requires increasing the number of workers by a factor of 3 Oneninth as many workers are needed Onethird as many workers are needed The same number of workers are needed Three times as many workers are needed Nine times as many workers are needed Part F The loudness of a sound is inversely proportional to the square of your distance from the source of the sound If your friend is right next to the speakers at a loud concert and you are four times as far away from the speakers, how does the loudness of the music at your position compare to the loudness at your friend's position? The sound is onesixteenth as loud at your position The sound is onefourth as loud at your position The sound is equally loud at your position The sound is four times as loud at your position The sound is sixteen times as loud at your position Inversesquare relations show up in the loudness of sounds, the brightness of lights, and the strength of forces
29 Proportional reasoning is useful for checking your answers to problems If your answer is a formula, then you can explicitly check that all of the variables have the correct proportionalities If you have a numerical answer, you can check your technique by doubling one of the starting variables and working through the same process to a solution If your answer does not change as you expect it to based on the proportionality of the initial and final variables, then you know that something is wrong
30 Optimizing the Bakery Learning Goal: To use substitution and cancellation to solve a system of two linear equations in two unknowns Frequently, in physics, you will have two different unknown quantities (for instance, distance and time) If you have two different equations involving these two unknown quantities, then you will be able to find exact values for each of them These will usually consist of two linear equations equations that only contain terms like 2x, y, and constant numbers with two unknowns In this problem, you will learn how to solve such systems, as well as see how equations of this type can be used to make a business more profitable Suppose that a baker makes cakes and cookies He knows that he is most efficient when he makes pairs of cakes (instead of one cake at a time) or a dozen dozens of cookies Call the number of pairs of cakes that he bakes in a week x and the number of grosses (dozens of dozens) of cookies that he bakes in a week y There are two factors that limit how much he can bake in a week: He only wants to work for 40 hours a week and he only has one oven Suppose that it takes the baker 1 hour to prepare a pair of cakes or a gross of cookies (before they are placed in the oven) Since he only wants to work 40 hours a week, his output of pairs of cakes x and his output of grosses of cookies y are constrained by the equation x + y =40 To maximize the profit of the bakery, the first step is to find where the equations for all of the constraints intersect For the following part, you will look at x + y=40 and y=0, which is also a constraint (specifically a minimum) since the baker cannot make a negative number of cookies Parts A and B might seem easier than most problems with linear systems, but in them you will use the basic techniques needed to solve any linear system: adding equations to cancel variables and substituting the value of one variable to find the value of the other Part A One way of solving systems of linear equations is by adding a multiple of one equation to the other The multiplier for the first equation is chosen so that one of the two variables will cancel out in the sum What should you multiply the equation y=0 by so that when added to x + y = 40 the variable y will cancel out? Express your answer numerically Hint 1 Consider another example Suppose that you are trying to solve the system 5x + 2y = 10 and x + y =3, and you wish to cancel the variable y In the first equation you have the term 2y You can cancel this term by adding the inverse, namely, 2y How can you get a 2y term in the second equation? Since the y term in the second equation is simply y, you must multiply the second equation through by 2 This changes the second equation to 2x 2y = 6 When you add this to original equation, the variable y will not appear in the sum In this part, you are trying to cancel the term y from the first part What is the inverse of y?
31 1 This leads to the sum Thus, the point of intersection for the lines y =0 and x + y = 40 is (40,0) This point tells you that if the baker makes no cookies, he can make (at most) 40 pairs of cakes in a week Part B Which of the following shows the result of substituting x = 0 into 075x + 15y = 40? Hint 1 How to approach the problem Once you have a numeric value for one of the variables, it is usually easier to substitute that value into one of the original equations and then solve for the other variable In this case, you know the value of x from the second equation Replacing the x in the first equation with a zero gives 075x + 15y = 40 After simplifying this equation, you can solve it to find the value of y Dividing both sides of this equation by 15 gives y = 40/15 = 80/3 This point tells you that, if the baker makes no cakes, he can bake (at most) 267 grosses of cookies Part C Next, you need to find the x coordinate of the point of intersection for the equations x + y =40 and 075x + 15y = 40 Multiply x = y = 40 by a number that will make the y s cancel when you add the two equations Which of the following gives the resulting equation? When multiplying an equation so that you can cancel one of the variables, be certain to multiply both sides of the equation For instance, multiplying the above equation incorrectly would yield: 15x 15y = 40, which would give you very incorrect values when you tried to solve for x and y Part D Now, add the two equations and find the value of x Express your answer to three significant figures x= 267 Part E Finally, substitute the value 267 (or the exact value of 80/3) for x in the first of the original equations (x + y = 40) to find the value of y Once you have found the value of y using the first of the original equations, you can substitute this value of y and the value of x(267) into the second of the original equations (075x + 15y =
32 40) to check your work If the values that you find on the two sides of the equal sign are not equal, then you need to recheck your work Express your answer to three significant figures = 133 Suppose that this baker makes a profit of $20 on each pair of cakes and a profit of $25 on each gross of cookies Then his profit P is given by the formula P=20x +25y To maximize a linear function (such as this profit function) given linear constraints (such as those given in the beginning of the problem) you only need to look at the points of intersection of the constraints In this case, the four points to check are (0,0), (40,0), (0,267), and (267,133) Plug the x and y values from each point into the profit function to find which gives the greatest profit Part F Find the solution of the system of equations 3x + 4y = 10 and x y =1 Give the x value followed by the y value, separated by a comma Express your answer in the form x,y Hint 1 Find the x value of the solution Multiply the second equation by a number that will make the y terms cancel when the equation is added to the first equation Then solve for x in the resulting equation Express your answer as an integer Hint 1 Find the proper multiplier for x y = 1 You wish to cancel the 4y term from the first equation when you add the two equations (after multiplying the second by some constant) The inverse of 4y is 4y What do you need to multiply y by to get a 4y term in the second equation? Express your answer as an integer 4 Now, find the sum of the equations 3x + 4y = 10 and 4x 4y = 4 and solve for x in the resulting equation x = 2 Now, substitute 2 for x in one of the two original equations and solve for y in the resulting equation x,y= 2, 1
33 Pay Up! Learning Goal: To learn to solve linear equations Almost every topic in physics will require you to solve linear equations equations that don't contain any higher powers of the variable such as x 2, x 3, etc Linear equations are the simplest algebraic equations They arise in all sorts of situations For this problem we'll look at one that might come up in your daily life Suppose that you and three friends go out to eat and afterward decide to split the cost evenly Your friend Anika points out that she only had a drink, so she should pay less ($2, the cost of her drink) and the rest of you can split the remainder of the bill A linear equation can easily determine how much each of you must pay For the particular problem raised in the introduction, assume that the total bill is $44 To answer the question "How should the bill be split?" we will create a linear equation The unknown is how much money a single person (besides Anika) must pay, so call that x Although four people (you plus three friends) went to dinner, only three are paying the unknown amount x for a total of 3x Since Anika is paying $2, the total amount paid is 3x + 2 dollars, which must equal the amount of the bill, $44 Thus, the equation to find is 3x + 2 = 44 The steps for solving a linear equation are as follows: 1 Move all of the constants to the right side 2 Move all of the variable terms (terms containing ) to the left side 3 Divide both sides by the coefficient of the variable to isolate the variable You will go through these steps one at a time to solve the equation and determine how much each person should pay Part A The first step in solving a linear equation is moving all of the constants (ie, numbers like 2 and 44 that aren't attached to an x) to the right side What is the final value on the right side once you've moved all of the constants? Express your answer as an integer Hint 1 How to "move" a constant The term move is not exactly an accurate description of how you get rid of the constant on one side of an equation What you actually do is add the opposite of that number to both sides In this way, the constant no longer appears on the side that you don't want it on For instance, if you have x + 3 =4, you want to move the 3 to the right The opposite of 3 is 3, so you would subtract 3 from both sides:,
34 which reduces to Similarly, if you had 2x 12 = 3, you would add 12 to each side, because the opposite of 12 is 12, and so, 3x= 42 Part B Now that you have 3x = 42, you need to isolate the variable so that you have an equation of the form "x= some number" What is the value of x (ie, the amount you must pay)? Express your answer as an integer number of dollars Hint 1 How to isolate the variable To remove the coefficient from the variable, simply divide both sides of the equation by that coefficient For instance, if you had 2x = 6, then you would divide both sides by 2: yielding, x= 14 The next problem looks more intimidating, but it requires the same procedures: Move all constants to the right, then move all variables to the left, and finally divide both sides by the variable s coefficient Part C If 13x 23 = 4x + 22, what is the value of x? Express your answer as an integer Hint 1 Collect the constant terms You have the equation 13x 23= 4x + 22 and need to get all of the constants on the right side Which of the following would get all of the constant terms on the right side of the equation? Add 23 to both sides
35 Subtract 23 from both sides Add 22 to both sides Subtract 22 from both sides Hint 2 Collect the terms with x After adding 23 to both sides, you have the equation 13x = 4x + 45 and need to get all of the terms with x on the left side Which of the following would get all of the variable terms on the left side? Add 13x to both sides Subtract 13x from both sides Add 4x to both sides Subtract 4x from both sides After subtracting 4x from both sides of the equation, you are left with 9x = 45 Dividing both sides by the coefficient (9) of x will give you the value of x Remember that you must always add and subtract to get all of the constant terms on the right side and all of the variable terms on the left side before multiplying or dividing to isolate the variable x= 5 Part D If 5x 9 = 2x 19, what is the value of x? Express your answer as an integer Hint 1 Collect the constant terms You have the equation 5x 9 = 2x 19 and need to get all of the constants on the right side Which of the following would get all of the constant terms on the right side of the equation? Add 9 to both sides Subtract 9 from both sides Add 19 to both sides Subtract 19 from both sides Hint 2 Collect the terms with x After subtracting 9 from both sides, you have the equation 5x= 2x 28You need to get all of the terms with x on the left side Which of the following would get all of the variable terms on the left side?
36 Add 5x to both sides Subtract 5x from both sides Add 2x to both sides Subtract 2x from both sides After adding 2x to both sides of the equation, you are left with 7x = 28 Dividing both sides by the coefficient (7) of x will give you the value of x Remember that you must always add and subtract to get all of the constant terms on the right side and all of the variable terms on the left side before multiplying or dividing to isolate the variable x= 4
37 Solving Linear Equations Learning Goal: To learn how to solve and check linear equations in one unknown Algebra may seem puzzling, but this problem will show that solving linear equations comes down to arithmetic Before trying to solve an equation, you would like to have an idea of what the solution should be, so that the solution can be checked after solving An equation can always be turned into a question in words, which highlights the arithmetic nature of the problem Rephrasing an equation in words also makes it easier to estimate the value of the solution For instance, solving the equation 4x + 8 =60 is the same as answering the question "What number can you multiply by 4 and then add 8 to the product to get 60?" Part A Consider the numbers 0, 10, 20, 30, and 40 Multiply each by 4 and compare the result to 60 to determine into which of the following intervals the answer to the question "What number can you multiply by 4 and then add 8 to the product to get 60?" should fall Select the interval into which the answer should fall 0 to to to to 40 Part B The answer here, x = 7, is not in the interval that you selected in the previous part What is wrong with the work shown above? In the first step, both terms on the left should be divided by 4 60/4 should yield 16, not 15 The last step should be to add 8 rather than subtract 8 The last step should be to subtract 15 from each side Remember that when solving equations you are performing actions on both sides of the equality, not on individual elements on each side Think "Subtract 2 from both sides" rather than "I'll subtract 2 from here, and from over here" The distinction isn't obvious when you add or subtract,
38 but it is with multiplying and dividing Notice "I'll divide both sides by 4" implies the correct action: dividing each term on both sides by 4 If you just think "I need to divide here and here," you are more likely to make errors such as the one just demonstrated Part C Recall that in Part A the equation 4x + 8 =60 was restated in words as, "What number multiplied by 4 and then added to 8 gives 60?" Which of the following would be an equivalent way to write this? What number can you multiply by 4 to get a number 8 less than 60? What number can you subtract 60 from and then divide by 4 to get 8? What number can you multiply by 4 to get a number 8 larger than 60? What number can you divide by 4 to get a number 8 less than 60? Part D You know that a number 8 less than 60 is the number 52 So, you could rewrite your answer from the previous part as, "What number can you multiply by 4 to get 52?" This is just the definition of 52 divided by 4! So, what is the number x you are looking for? Express your answer as an integer x = 13 You've successfully solved this equation using simple arithmetic, and your answer falls into the interval that you selected in Part A What you just did was the same as following more formal algebraic operations Namely, Part C corresponds to the step and Part D corresponds to the step When you run into algebraic equations to solve, instead of trying to remember a list of rules, just think about the arithmetic Then, you will naturally arrive at the correct steps to manipulate the equation to find a solution for x
39 Solving Quadratic Equations Learning Goal: To use the quadratic formula to solve quadratic equations Maria wants to plant a small tomato garden in her yard She bought 25 tomato plants, and she has read that ideally tomatoes are planted in a square grid to help them pollinate each other Part A How many plants x should she plant in each row so that her 25 plants end up in a square (ie, x plants in each of x rows)? Express your answer as an integer Hint 1 How to approach the problem If the gardener has x plants in each of her x rows, then the total number of plants will be x 2 Since there are 25 total plants, the correct value of x will satisfy the equation x 2 = 25 Solve this equation by taking the square root of both sides Note that 25 has both a positive and a negative square root Since you can't have a negative number of plants, you only want the positive square root 5 You may have been able to solve this part simply by intuition or with simple arithmetic In the next part, you will use the quadratic formula to find the value of x Although this involves more work than necessary to solve this part, using the quadratic formula on a problem that you've already solved should help you to feel comfortable with it Part B To find x in Part A, you would need to solve the equation x 2 25 =0 Which of the following shows the proper values in the quadratic formula before simplifying the radical and dividing? Hint 1 The values of a, b, and c A is the value of the coefficient of x 2, so in this problem, a = 1 (because x 2 = 1*x 2 ) bis the value of the coefficient of x Notice that the equation given has no x term Since 0*x=0, the coefficient of x must be zero Therefore, b=0 Finally, c is the constant term, so in this problem, c = 25 Plug these values into the quadratic formula and then reduce it to the form shown in one of the answer choices
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