Volume and Surface Area of a Sphere

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1 Volume and Surface rea of a Sphere Reteaching 111 Math ourse, Lesson 111 The relationship between the volume of a cylinder, the volume of a cone, and the volume of a sphere is a special one. If the heights and diameters of a cylinder and a cone equal the diameter of a sphere, then: The volume of a cone is 1 the volume of a cylinder, and The volume of the sphere is twice the volume of the cone, and 2 the volume of the cylinder. ecause of this special relationship, we are able to determine the formula for the volume of a sphere: Formula for the Volume of a Sphere 4 πr The surface area of a sphere is 4 times the area of its greatest cross-sectional area. It s surface area also equals the lateral surface area of a cylinder with a diameter and height equal to the diameter of the sphere. Formula for the Surface rea of a Sphere = 4πr 2 1. What fraction of the volume of a cylinder is the volume of a cone when both objects have the same height and base diameter? 2. What fraction of the volume of a cylinder is the volume of a sphere when the height and diameter of the cylinder equals the diameter of the sphere?. Find the volume and surface area of a 12-inch diameter playground ball. (Express in terms of π.) Volume rea 4. Find the volume of a sphere with a radius of 6 cm. Round to the nearest ten cubic centimeters. (You may use a calculator and use.14 for π.) 122 Harcourt chieve Inc. and Stephen Hake. ll rights reserved. Saxon Math ourse

2 Ratios of Side Lengths of Right Triangles Reteaching 112 Math ourse, Lesson 112 Trigonometry is a branch of mathematics that explores the relationships between the angles and sides of triangles. Some special math language is used in trigonometry. Opposite Side and djacent Side For each acute angle in a right triangle, one of the legs is the opposite side and the other leg is the adjacent side. From the perspective of, is the opposite side and is the adjacent side. From the perspective of, is the opposite side and is the adjacent side. We can find the ratios of side lengths. The ratio of the lengths of the opposite side of to the hypotenuse is: opposite hypotenuse = 4 = 0. The side opposite X is YX. The hypotenuse is XY. The ratio of the side opposite to the hypotenuse is: opposite hypotenuse = 12 = Find each ratio for the illustrated right triangle. Express each ratio as a fraction and as a decimal rounded to three decimal places. 4 With respect to, find the ratio of the lengths: Z X 12 Y a. Opposite side to adjacent side b. Opposite side to hypotenuse c. djacent side to hypotenuse 2. With respect to, find the ratio of the lengths: a. Opposite side to adjacent side b. Opposite side to hypotenuse c. djacent side to hypotenuse Saxon Math ourse Harcourt chieve Inc. and Stephen Hake. ll rights reserved. 12

3 Using Scatterplots to Make Predictions Reteaching 11 Math ourse, Lesson 11 Scatterplots are graphs of points representing a relationship between two variables. elow are three scatterplots that display some common relationships. This scatterplot indicates a strong relationship between the variables plotted. It has a positive correlation, which means that as one variable increases, the other variable increases. This scatterplot also indicates a strong relationship; however, it has a negative correlation. s one variable increases, the other decreases. This scatterplot shows no meaningful relationship. Researchers use these terms to describe relationships between variables. They also describe these relationships algebraically to determine the predictability of the relationship. In this scatterplot we see that the line of best fit passes through the origin and point (10.0, 27.0), so it s slope is 2.7. Therefore, the algebraic equation is: y = 2.7x In this equation, x represents volume and y represents mass. This equation can be used to predict the mass if the volume is 100 cm : y = 2.7x; y = ; y = 270 g 1. Identify each graph as having a positive correlation, a negative correlation, or no meaningful relationship. a b c 2. company increased its advertising minutes over the course of a year. The advertising minutes and the sales in are recorded in the scatterplot. Draw a line of best fit. Then predict the dollars in sales when the advertising minutes reach Harcourt chieve Inc. and Stephen Hake. ll rights reserved. Saxon Math ourse

4 alculating rea as a Sweep Reteaching 114 Math ourse, Lesson 114 Refer to Lesson 114 for guidance to solve the following problems. 1. Segment is 10 cm and the diameter of the cylinder is 12 cm. Describe how to calculate the lateral surface area of the cylinder. Find the lateral surface area expressed in terms of π. 2. Imagine slant height segment sweeping once around the cone. The points on move an average of half the circumference. Find the lateral surface area of the cone in terms of π. 10 cm cm. Imagine a broom steadily narrowing from 14 in. to 10 in. as it sweeps inches to form a trapezoid. Multiply the average length of broom times the distance swept, to find the area of the trapezoid. 4. sector of a circle is used to create the lateral surface of a cone with a slant height of 6 inches and a diameter of 4 inches. Using for π, estimate the lateral surface area of the cone. Saxon Math ourse Harcourt chieve Inc. and Stephen Hake. ll rights reserved. 12

5 Relative Sizes of Sides and ngles of a Triangle Notice the relationships between angle size and sides of these triangles: m < m < m The side opposite the smallest angle is the shortest side. The side opposite the largest angle is the longest side. Reteaching 11 Math ourse, Lesson 11 The order of side lengths of a triangle, when written from shortest to longest, corresponds to the order of the sides opposite angles when written from smallest to largest. m < m < m < < 1. Place the angles in order from smallest to largest Place the sides in order from shortest to longest Two sides of a triangle are equal in measure and are greater than 60. Sketch a triangle that fits this description Harcourt chieve Inc. and Stephen Hake. ll rights reserved. Saxon Math ourse

6 Division by Zero Reteaching 116 Math ourse, Lesson 116 Using a calculator, try to divide 0. Try dividing any number by 0. 0 Notice that the calculator displays an error message when division by zero is entered. The display is frozen and other calculations cannot be performed until the erroneous entry is cleared. We say that division by zero is undefined. When performing algebraic operations, we guard against dividing by zero. 1. Using a calculator, divide each of these numbers by 0: 7, 4, 104, 4, 222. Remember to clear the calculator between calculations. What answers are displayed? 2. If we attempt to form two division facts for the multiplication fact 9 0 = 0, one of the arrangements is not a fact. Which arrangement is not a fact and why?. Under what circumstances does the following expression not equal 1? Why? x 7 x 7 4. Find the number or numbers that may not be used in place of the variables in these expressions: a. 12 x b. x c. x 4 d. y + 7 y 9 Saxon Math ourse Harcourt chieve Inc. and Stephen Hake. ll rights reserved. 127

7 Significant Digits Reteaching 117 Math ourse, Lesson 117 ecause measures are not exact, a measure can be expressed only to a degree of precision. The number of digits we use to express a measure with confidence is called the number of significant digits. elow are rules for counting significant digits: 1. ount all non-zero digits. (The number 4.7 has significant digits.) 2. ount zeros between non-zero digits. (The number 20. has significant digits.). ount trailing zeros in a decimal number. (The number 2.40 has significant digits.) 4. Do not count zeros if there is not a non-zero digit to the left. (The number 0.04 has 2 significant digits.). Trailing zeros of whole numbers are ambiguous. The number of significant digits in 100 might be 2, or 4. Expressing a number in scientific notation removes the ambiguity. (The number has significant digits.) 6. Exact numbers have an unlimited number of significant figures, so we do not count any digits when dealing with exact numbers. Exact numbers include counts and exact conversion rates like 12 in./ft and 2.4 cm/in. Follow these two rules when performing calculations with measures: product or quotient of two measurements may only have as many significant digits as the measurement with the least number of significant digits. sum or difference may show no more decimal places than the measurement with the fewest number of decimal places. 1. Write the significant digits in the following numbers. a min b cm c ml d g 2. Perform each calculation and express each answer with the correct number of significant digits. a. 12. hrs 1. min b. 2.7 lbs +.9 lbs + 2. lbs 12 Harcourt chieve Inc. and Stephen Hake. ll rights reserved. Saxon Math ourse

8 Sine, osine, Tangent Reteaching 11 Math ourse, Lesson 11 Trigonometry Ratios Name bbreviation Ratio sine sin opposite hypotenuse cosine cos adjacent hypotenuse tangent tan opposite adjacent You can use a calculator to find the decimal values of trig ratios. On a graphing calculator the trig functions are often abbreviated. 1. Name the trig function for each ratio. a. opposite hypotenuse b. adjacent hypotenuse c. opposite adjacent 2. Find the sine, the cosine, and the tangent for T. Express each ratio as a fraction and as a decimal. cm E M 4 cm cm T. Use a calculator with trig functions to find to the nearest thousandth: 2 24 a. the sine of. b. the tangent of. c. the cosine of. Saxon Math ourse Harcourt chieve Inc. and Stephen Hake. ll rights reserved. 129

9 omplex Fractions Reteaching 119 Math ourse, Lesson 119 ll of the above fractions are examples of complex fractions. complex fraction is a fraction containing one or more fractions in the numerator or denominator. It is customary to express fractions with integers when possible. elow are two methods for simplifying a complex fraction: Method One Simplify 2 Step One: Find a common denominator: 20. Step Two: Use the Identity Property for Multiplication. Multiply the complex fraction by : Method Two Simplify = 2 1 Step One: Treat the complex fraction as a fraction division problem: 4 Dividend Divisor Step Two: Solve = = 1. Simplify. 2 a. c b d Harcourt chieve Inc. and Stephen Hake. ll rights reserved. Saxon Math ourse

10 Rationalizing a Denominator Reteaching 120 Math ourse, Lesson When a fraction s denominator contains a radical, we take steps to form an equivalent fraction that has a rational number denominator. Follow these steps to rationalize the denominator in the fraction Use the identify property of multiplication to multiply the fraction by a fraction equivalent to 1. = 1 Multiplied Rationalized denominator e sure you reduce the fraction after you rationalize the denominator if possible Reduced Fraction with a radical Identity property of multiplication Rationalized denominator. 1. Rationalize the denominator of each fraction. a. 7 2 b. c. _ 10 e. 2 2 d. f Saxon Math ourse Harcourt chieve Inc. and Stephen Hake. ll rights reserved. 11

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