1ACE Exercise 3. Name Date Class

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1 1ACE Exercise 3 Investigation 1 3. A rectangular pool is L feet long and W feet wide. A tiler creates a border by placing 1-foot square tiles along the edges of the pool and triangular tiles on the corners, as shown. The tiler makes the triangular tiles by cutting the square tiles in half along a diagonal. 1ft W 1ft border tile L a. Suppose the pool is 30 feet long and 0 feet wide. How many square tiles does the tiler need for the border? 0 30 How many tiles are needed to make the two sides that are 30 feet? How many tiles are needed to make the two sides that are 0 feet? 190

2 1ACE Exercise 3 (continued) Investigation 1 How many tiles are needed to make the four corners? How many square tiles are needed to make the entire border? b. Write two equations for the number of square tiles N needed to make this type of border for a pool L feet long and W feet wide. 1.) N =.) N = c. Explain why your two equations are equivalent. 191

3 ACE Exercise 1 Option 1 Investigation 1. The student council is sponsoring a T-shirt sale to raise money for a local charity. They make the following estimates of expenses and income: Expense of $50 for advertising Expense of $4.5 for each T-shirt Income of $1 for each T-shirt Income of $150 from a business sponsor a. Write an equation for the income I made for selling n T-shirts. HINT Remember to include the income from the business sponsor. a. I = b. Write an equation for the expenses E for making n T-shirts. a. E = 19

4 ACE Exercise 1 Option 1 (continued) Investigation c. Suppose the student council sells 100 T-shirts (n = 100). What is their profit? How much income (I) will the student council have from selling 100 T-shirts (n = 100)? How much expense (E) will the student council have from making 100 T-shirts (n = 100)? If profit (P) is the difference between income (I) and expense (E), what is the profit if the student council sells 100 T-shirts? d. Write an equation for the profit P made from selling n T-shirts. a. P = HINT Combine the equations for income and expense to show profit. 193

5 ACE Exercise 1 Option Investigation 1. The student council is sponsoring a T-shirt sale to raise money for a local charity. They make the following estimates of expenses and income: Expense of $50 for advertising Expense of $4.5 for each T-shirt Income of $1 for each T-shirt Income of $150 from a business sponsor a. Write an equation for the income I made for selling n T-shirts. HINT Remember to include the income from the business sponsor. How much income (I) from printing the business s logo on the back of T-shirts does the student council make no matter how many T-shirts they sell? How much income (I) does the student council make from each T-shirt sold? What is the equation for income (I) if the student council sells n T-shirts? b. Write an equation for the expenses E for making n T-shirts. How much expense from advertising (E) does the student council have no matter how many T-shirts they sell? How much expense (E) does the student council have from each T-shirt made? What is the equation for expense (E) if the student council makes n T-shirts? 194

6 ACE Exercise 1 Option (continued) Investigation c. Suppose the student council sells 100 T-shirts (n = 100). What is their profit? c. How much income (I) will the student council have from selling 100 T-shirts (n = 100)? c. How much expense (E) will the student council have from making 100 T-shirts (n = 100)? c. If profit (P) is the difference between income (I) and expense (E), what will the profit be if the student council sells 100 T-shirts? d. Write an equation for the profit P made from selling n T-shirts. c. P = HINT Combine the equations for income and expense to show profit. 195

7 3ACE Exercise 9 Investigation 3 9. The school choir from Problem 3.1 has the profit plan P = 5s (100 + s). The school band also sells greeting cards. The equation for the band s profit is P = 4s (10 + s). Find the number of boxes s that each group must sell to have equal profits [P(choir) = P(band)]. HINT It will be helpful to show your work at every step. If the two plans are equal, then 5s (100 + s) = 4s (10 + s). To find the solution, get both sides of the equations in simplest terms: b. Distribute any numbers or negatives in front of parentheses. b. Add like terms. b. Get like terms on one side of the equation. b. Solve for number of boxes of greeting cards by getting 1s on one side by itself. b. Check your answer. Does the value you found for s result in the same profit (P) for the choir and the band? 196

8 4ACE Exercise (continued) Investigation 4. A new pump is used to empty the pool in Exercise 1. The equation w = 75(t 7) represents the gallons of water w that remain in the pool t hours after pumping starts. HINT It may help to write an equation that is equivalent to w = 75(t 7). w =. a. How many gallons of water are pumped out each hour? a. How many gallons of water are left after 1 hour? a. How many gallons of water are left after hours? a. How many gallons of water are pumped out each hour? b. How much water is in the pool at the start of the pumping (when t = 0)? b. w = 197

9 4ACE Exercise (continued) Investigation 4 c. Suppose there are 1,000 gallons of water (w = 1,000) left in the pool. How long has the pump been running? c. t = d. After how many hours will the pool be empty (w = 0)? c. t = e. Write an equation that is equivalent to w = 75(t 7). What information does it tell you about the situation? 198

10 5ACE Exercises 1 3 Investigation 5 Maria presents several number puzzles to her friends. She asks them to think of a number and to perform various operations on it. She then predicts the result. For Exercises 1 and, show why the puzzles work. Exercise 1 has been done for you as an example. Please do Exercise in a similar way. 1. Puzzle 1 Example: Pick a number. x is my number. Double it. I get x when I double it. Add 6. When I add 6, I get x + 6. Divide by. Dividing by, I get. Subtract the number you Subtracting x results in [ ] x. When I simplify, thought of. I get = x and = 3. So, I have (x + 3) x. x 6 x+6 x+6 Combining like terms, I get 3. The result is 3 for any number x I pick. Marie claims the result is 3.. Puzzle Think of a number. Add 4. Multiply by. Subtract 6. Divide by. Subtract the number you thought of. Marie claims the result is

11 5ACE Exercises 1 3 (continued) Investigation 5 3. a. Design a puzzle similar to Marie s puzzles. b. Try it on a friend. c. Explain why your puzzle works. 00

12 Unit Test 1. The Morales family is remodeling their home. The wall between the living room and dining room is going to be removed to make one big living space. Write two equivalent expressions for the area of the new living space. 0 x + 5 living room x dining room. Three of the following expressions are equivalent. Choose the expression that is not equivalent to the others and explain how you can tell, without using a calculator, that it is not equivalent. A. 8x 1x 4 B. 1x 16x 4 C. 4 4x D. 4(1 4x) HINT Write each expression in its simplest form. For example:. 8x 1x 4 4 4x.. Rewrite the other expressions (in their simplest form) and then decide which is not equivalent (the same). B. C. D. 01

13 Unit Test (continued) 3. For each expression below, write an equivalent expression. a. (15 4x) (10 x) b. (10 x) (30 3x) 4. Explain how you can tell, without using a calculator, that these expressions are not equivalent x 4x 5 4x(x 5) 5. a. Choose the expressions that represent the perimeter (distance around the figure) of the shape below. HINT There is more than 1 expression A. r r πr r r that represents the perimeter. B. 4r πr C. r πr D. r ( π) E. r (4 π) b. Explain why the expressions you chose are correct. r r 0

14 Unit Test (continued) 6. Solve the following equations. Show your work. a. (x 4)(x 6) = 0 b. 3(x 10) 5(x ) = 0 c. x x 0 = 0 d. (7x 15) = 18 x 03

15 Unit Test (continued) In 7 10, match each equation with two other equivalent representations. Exercise 7 has been done for you as an example. 7. y = x (x + 3) y = x x + x 3 y = x + 3x (C) A. y = x (3 x) B. y = x + x 8 C. y = x + 3x D. 8. y = 3x x E. 9. y = x 4x 10. y = (x )(x + 4) F. x y G H. x y x y