Algebra: Real World Applications and Problems

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1 Algebra: Real World Applications and Problems Algebra is boring. Right? Hopefully not. Algebra has no applications in the real world. Wrong. Absolutely wrong. I hope to show this in the following document. We ll cover topics like forming and solving equations and show where we might use this outside of the classroom. We ll move onto quadratics and spend a little while looking at applications of quadratics and parabolas. On the way we will try to cover extracting common factors along with the difference of two squares and show why these might be useful outside the classroom. Solving Equations We begin with the basics, that is solving simple equations. The most important thing to remember when solving equations is that whatever you do to one side you need to also do to the other. We will use letters x, y and z to denote the things that we want to find out. Take an example. Let x be the cost of a meal in a particular restaurant. Bob pays for two of these meals and the cost is 30. How much is each meal? 1

2 Too easy right. I didn t need algebra for that you re saying. Well ok maybe not but you can use it to help you out. How about this one? Bob goes to a different restaurant and orders the same two meals, but with drinks this time. The drinks cost 4 (cheap I know) and the total bill comes to 28. If y represents the cost of the meal, form an equation for y and solve it to find y. Still too straight forward or getting more complicated? Final example Bob goes to a third restaurant and his total bill is identical to another customer s, Cindy. Bob s order is two identical main courses, a 2 drink and a 4.50 pudding with his meal whilst Cindy just has one main course (the same one as Bob) and worth of drinks on her bill. If z represents the cost of the main meal both Bob and Cindy order, form an equation to solve for z and solve it. 2

3 Try these questions: i) Customers at a bar want to compare the price of a particular cocktail. Wendy got a 1 reduction on each of the 9 cocktails she ordered but her bill was the same as Jane s who bought only one of the cocktails but also got 15 of other drinks. (a) If we let x be the cost of the cocktail, form an equation for x. (b) Solve the equation to find x. ii) A supplier offers a discount on a product bought in bulk. If you buy 240 of them you get 20 off each item and this reduces the total cost to the same as buying 200 at their full price. (a) If we let x be the cost of the product, form an equation for x. (b) Solve the equation to find x. iii) Two competing firms offer a particular item at a reduced rate. The reductions from each firm are different. If you buy 360 of them from firm A, you get a 4 reduction off each item. For the same total price firm B gives you 320 of the same item but with a discount of 3 per item. (a) If we let c be the cost of the item, form an equation for c. (b) Solve the equation to find c. iv) Can you think of a question similar to those above that could result in the equation 23(x 1) = 5(x + 3)? Solve the equation for x. v) Produce a question that might result in the equation and solve the equation to find s. 14(s 7) + 2(s + 21) = 0 3

4 Ok now I ve got a real brain teaser for you. Bob would like you to work out his age. He gives you the following clue: One third of my age 8 years ago is equal to one fifth of how old I will be 5 years time. How old am I? How can we find his age? Using logic and algebra of course. 4

5 Here are some more problems for you to solve i) One sixth of my age three years from now is the same as one seventh of my age four years from now or if we let x be my age then Find x. x = x Similar word problems give rise to these equations. them? Can you solve ii) iii) iv) x x x 9 3 = x = x = x

6 Some more problems (optional). i) Using the equation V = Al (volume = area length) find l if A = 7cm 2 and V = 24.5cm 3. ii) Using the equation V = IR (voltage = current resistance) find the current I if the voltage is 13 volts and the resistance is 39 ohms. iii) Using the equation P = I 2 R (Power = current 2 resistance) find the current I if the power is 200 Watts and the resistance is 8 ohms. iv) Using the equation v = u + at (current velocity = original velocity + acceleration time) find the time t if v = 20m/s, u = 10m/s and a = 2m/s 2. v) Using the equation V = πr 2 h (which you should remember from the volume topic), find an expression for h and then find the value of h if V = 62m 3 and R = 3m. 6

7 Multiplying Brackets Let us move on. Farmer Giles has a field that is 20m wide and was originally 70m long. He increases the length of the field by x metres so that the total area of the field is now 3200m 2. The picture below shows his field. How much has he increased the length of his field by? 7

8 A more complicated example. The farmer s cousin, Esmeralda, has a kitchen that is 3m wide by 8m long. She is getting an extension and the architect wants a general equation to show the area of her new kitchen. Due to planning restrictions the architect must increase the length and depth by the same amount. The architect draws the diagram below to help him. What is the general equation for the area of the new kitchen? 8

9 Find the equations of the areas of these rooms and find the area of these photo frames. (Hint: Find the area of the outside rectangle and subtract the area of the inside rectangle. 9

10 Now you can find the area of these shapes and finally find the volume of this cuboid. If you wish to do more practice (remember you get tested on the less interesting routine multiplying and rearranging of these brackets) you can use exercise 2 on page 50 of the textbook. 10

11 Factorising Now we will move on from multiplying things together to pulling them apart, little by little. We will talk about common factors and multiplying quadratics. We begin with common factors. Common Factors Recognising common factors makes simplifying equations easier. What use is this you might say? Well you go to the supermarket to buy 4 apples, 4 bananas, 4 boxes of strawberrys and 4 boxes of blueberrys to make smoothies. You want to make sure you ve got enough money to pay for everything (and you ve forgotten anything that might have the ability to be a calculator before you suggest that). Is it going to be easier to do i) 4 apples + 4 bananas + 4 blueberries + 4 strawberries or ii) 4 ( apples + bananas + blueberries + strawberries)? You will get the same answer no matter which you do and as you re only multiplying by 4 it might not make too much of a difference, what if you were multiplying by 23? I don t know about you but I would probably do ii). We can use common factors in more complicated equations/examples as well, we ll see if in all sorts of problems later including quadratic equations which is the next thing we ll cover. What does this sort of problem look like? Well something like this i) Simplify 21x + 14y + 28z. Which could be taken as, simplify the sum that gives us the cost of 21 packs of double A batteries, 14 nine volt batteries and 28 packs of triple A batteries. 11

12 Now for a more complicated problem. Simplify this expression for the area of the shape and try to work out possible expressions for its length and width. Try a selection of questions from exercise 4 on page 53 of your textbook. 12

13 Quadratic Equations First a bit of background which we will need this for later anyway. quadratic equation that looks like this A y = ax 2 + bx + c. We need to learn how to break one up (factorise) into separate brackets. Remember that (x + 1)(x + 1) = x 2 + 2x + 1, well we will learn how to go in the opposite direction x 2 + 2x + 1 = (x + 1)(x + 1) = (x + 1) 2. Why on earth might we want to do this? Well, if you plot a quadratic equation you get a graph that is called a parabola. The following pictures are examples of parabolas in the real world Projectile motion, suspension bridges, acceleration due to gravity all use quadratic equations. Why do we want to be able to factorise quadratics (split them up into two brackets)? Well say we are building a sculpture like the St Louis Arch shown above (middle picture). We know the equation needed to follow to make it the correct shape but we want to make sure we have enough room on the ground for it to fit into. Solving ax 2 + bx + c = (x A)(x B) = 0 by factorising the quadratic tells us where the new statue will hit the ground (at A and B). 13

14 So thats why we might want to factorise. How do we do it? Well its all about your times tables and a bit of trial and error. Let me show you some examples. We ll begin with something nice and simple. i) Factorise x 2 + 4x

15 ii) Let s do something a little more complicated. Factorise x 2 13x

16 iii) One more example (slightly different). Factorise x 2 5x 36. Try a selection of questions from exercise 6 on page 56 of your textbook. 16

17 There is another type of quadratic that you need to be able to factorise. That is one where you don t just have x 2, but a multiple of it. For example: iv) Factorise 4x x 12. Try a selection of questions from exercise 7 on page 57 of your textbook. 17

18 Difference of 2 Squares This is the final part of our algebra topic and it is a special case of the polynomial equations known as difference of 2 squares. If you can identify a polynomial as an example of this special case then factorisation is much easier. Take an example. We want to find the shaded area in this diagram 18

19 How about the difference between the area of these two shapes? Try a selection of questions from exercise 5 on page 54 of your textbook, including some parts of question 4. 19

20 Summary of Factorisation There is a three step process when it comes to solving factorisation problems in exams and homework. i) Common factor. ii) Difference of 2 squares. iii) Factorise. First you always check for a common factor it will make any other factorisation you have to do much easier. Secondly check if the expression is a difference of 2 squares because these are a special case. Finally, factorise. 20

21 Solutions to Problems Solving Equations Page 3 i) (a) 9(x 1) = x + 15 (b) x = 3. ii) (a) 240(x 20) = 200x (b) x = 120. iii) (a) 360(c 4) = 320(c 3) (b) c = 12. iv) x = 19 9 v) x = 7 2 Page 5 i) x = 3 ii) x = 2 iii) x = 22 iv) x = 48 Page 6 (optional questions) i) l = = 3.5m ii) I = = 1 3 amps iii) I 2 = = 25, I = 25 = 5amps iv) t = = 5secs v) h = V πr 2, h = 62 9π = 2.193m Page 9 1 (x + 1)(x + 7) = x(x + 7) + 1(x + 7) = x 2 + 7x + x + 7 = x 2 + 8x (x + 3)(x 5) = x(x 5) + 3(x 5) = x 2 5x + 3x 15 = x 2 2x 15 21

22 3 (x 2)(x 2) = x(x 2) 2(x 2) = x 2 2x 2x + 4 = x 2 4x (3 x)(x + 12) = 3(x + 12) x(x + 12) = 3x + 36 x 2 12x = 36 9x x 2 Q5 14x 12(x 4) = 14x 12x + 48 = 2x + 48 Q6 x(x+4) (x 3)(x 1) = x 2 +4x [x 2 4x+3] = x 2 +4x x 2 +4x 3 = 8x 3 Page 10 i) (x 1)(x 2 + 2x 3) = x(x 2 + 2x 3) 1(x 2 + 2x 3) = x 3 + 2x 2 3x x 2 2x + 3 = x 3 + x 2 5x + 3 ii) (x + 7)(3x 2 2x 1) = x(3x 2 2x 1) + 7(3x 2 2x 1) = 3x 3 2x 2 x + 21x 2 14x 7 = 3x x 2 15x 7 iii) (x 2 + 3)(x ) = x 2 (x ) + 3(x ) = x x 2 + 3x = x x The volume of the cuboid is given by (x + 1)(x + 4)(x 2) = (x + 1) [x(x 2) + 4(x 2)] = (x + 1) [ x 2 2x + 4x 8 ] = (x + 1) ( x 2 + 2x 8 ) = x(x 2 + 2x 8) + 1(x 2 + 2x 8) = x 3 + 2x 2 8x + x 2 + 2x 8 = x 3 + 3x 2 6x 8 22

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