Complex Expressions

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Answer Key & Explanations

Complex Expressions · Grade 9 · Worksheet 3

1. Noah is analyzing the cost of a new smartphone plan. The monthly cost C in dollars for using n gigabytes of data is modeled by the expression C(n) = 45 + 12n, where n represents the number of gigabytes used in a month. What does the number 12 represent in the context of this phone plan? Answer: The cost per gigabyte of data used. Solution: Identify the expression C(n) = 45 + 12n. The variable n is the number of gigabytes used. The term 12n means that for each gigabyte (each unit of n), the cost increases by 12 dollars.
   Full step-by-step solution

   Step 1: Identify the expression C(n) = 45 + 12n. The variable n is the number of gigabytes used. Step 2: The term 12n means that for each gigabyte (each unit of n), the cost increases by 12 dollars. Step 3: The constant term 45 is the fixed monthly fee, independent of data usage. Step 4: Therefore, the number 12 represents the cost per gigabyte of data, or the rate of change in cost with respect to data usage. The answer is: The cost per gigabyte of data used.

2. Hana is analyzing the expression 1200(1.08)^t that models her investment growth. In this expression, what does the number 1.08 represent? Answer: 1.08 represents the growth factor per time period, meaning the investment grows by 8% each period Solution: The expression is in the form A = P(1 + r)^t, where P is the principal amount, r is the growth rate, and t is time. In this case, 1.08 = 1 + 0.08, where 0.08 represents the growth rate.
   Full step-by-step solution

   Step 1: The expression is in the form A = P(1 + r)^t, where P is the principal amount, r is the growth rate, and t is time. Step 2: In this case, 1.08 = 1 + 0.08, where 0.08 represents the growth rate. Step 3: Converting 0.08 to a percentage gives 8%. Step 4: Therefore, 1.08 represents the growth factor, meaning the investment multiplies by 1.08 (or grows by 8%) each time period. The number 1.08 represents the growth factor per time period, indicating 8% growth.

3. Tane is analyzing the profit from selling handmade sculptures at a market. The profit P(x) in dollars for selling x sculptures is modeled by the expression P(x) = -2x^2 + 120x - 400. Interpret the meaning of the term -400 in this context. Answer: The term -400 represents the fixed costs or initial expenses (such as booth rental and materials) that Tane must pay regardless of how many sculptures are sold. Solution: The profit function is P(x) = -2x^2 + 120x - 400. Evaluate P(0) by substituting x = 0: P(0) = -2(0)^2 + 120(0) - 400 = 0 + 0 - 400 = -400.
   Full step-by-step solution

   Step 1: The profit function is P(x) = -2x^2 + 120x - 400. Step 2: Evaluate P(0) by substituting x = 0: P(0) = -2(0)^2 + 120(0) - 400 = 0 + 0 - 400 = -400. Step 3: When zero sculptures are sold, the profit is -400 dollars, meaning Tane has a loss of $400. Step 4: This loss represents the fixed costs that Tane must pay regardless of sales, such as booth rental, tools, or materials bought upfront. Step 5: Therefore, the term -400 in the expression represents the initial fixed costs or expenses that Tane incurs before any sales are made. The answer is: The term -400 represents the fixed costs or initial expenses (such as booth rental and materials) that Tane must pay regardless of how many sculptures are sold.

4. log₃(27) + 2⁴ - √(169) = ? Answer: 6 Solution: Evaluate log₃(27). Since 3^3 = 27, log₃(27) = 3. Evaluate 2⁴.
   Full step-by-step solution

   Step 1: Evaluate log₃(27). Since 3^3 = 27, log₃(27) = 3. Step 2: Evaluate 2⁴. 2 × 2 × 2 × 2 = 16. Step 3: Evaluate √(169). Since 13 × 13 = 169, √(169) = 13. Step 4: Substitute the values back into the expression: 3 + 16 - 13. Step 5: Perform the addition: 3 + 16 = 19. Step 6: Perform the subtraction: 19 - 13 = 6. The answer is 6.

5. Liam is designing a rectangular garden with a perimeter of 40 meters. He wants the length to be 4 meters more than the width. Write an equation in terms of width w that represents this situation, then determine the dimensions of the garden. Answer: width = 8 meters, length = 12 meters Solution: Define the variables. Let \( w \) = width of the garden (in meters). Let \( l \) = length of the garden (in meters).
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Define the variables.** Let \( w \) = width of the garden (in meters). Let \( l \) = length of the garden (in meters). --- **Step 2: Translate the given facts into equations.** We are told: 1. The perimeter is 40 meters. Formula for perimeter of a rectangle: \( P = 2l + 2w \) So: \( 2l + 2w = 40 \) 2. The length is 4 meters more than the width. So: \( l = w + 4 \) --- **Step 3: Substitute the second equation into the first.** From \( l = w + 4 \), substitute into \( 2l + 2w = 40 \): \( 2(w + 4) + 2w = 40 \) --- **Step 4: Simplify and solve for \( w \).** \( 2w + 8 + 2w = 40 \) \( 4w + 8 = 40 \) Subtract 8 from both sides: \( 4w = 32 \) Divide by 4: \( w = 8 \) --- **Step 5: Find \( l \).** \( l = w + 4 = 8 + 4 = 12 \) --- **Step 6: State the dimensions.** Width = 8 meters Length = 12 meters --- **Final Answer:** width = 8 meters, length = 12 meters

6. Emma is analyzing the trajectory of a water fountain using quadratic functions. The water's height above the ground is modeled by h(t) = -4.9t² + 14.7t + 2, where h is height in meters and t is time in seconds. At what time does the water reach its maximum height? Answer: 1.5 Solution: Identify the coefficients from the quadratic function h(t) = -4.9t² + 14.7t + 2 a = -4.9, b = 14.7, c = 2 For a quadratic function in standard form, the vertex (maximum point) occurs at t = -b/(2a) Substitute the values: t = -14.7/(2 × -4.9) Calculate the denominator: 2 × -4.9 = -9.8 Complete…
   Full step-by-step solution

   Step 1: Identify the coefficients from the quadratic function h(t) = -4.9t² + 14.7t + 2 a = -4.9, b = 14.7, c = 2 Step 2: For a quadratic function in standard form, the vertex (maximum point) occurs at t = -b/(2a) Step 3: Substitute the values: t = -14.7/(2 × -4.9) Step 4: Calculate the denominator: 2 × -4.9 = -9.8 Step 5: Complete the division: t = -14.7/(-9.8) = 1.5 Step 6: The water reaches its maximum height at t = 1.5 seconds The answer is 1.5.

7. log₂(8) + 3² - √(25) = ? Answer: 7 Solution: Evaluate log₂(8). Since 2³ = 8, log₂(8) = 3 Evaluate 3². 3 × 3 = 9 Evaluate √(25).
   Full step-by-step solution

   Step 1: Evaluate log₂(8). Since 2³ = 8, log₂(8) = 3 Step 2: Evaluate 3². 3 × 3 = 9 Step 3: Evaluate √(25). The square root of 25 is 5 Step 4: Substitute the values: 3 + 9 - 5 Step 5: Perform addition: 3 + 9 = 12 Step 6: Perform subtraction: 12 - 5 = 7 The answer is 7.

8. Sophia is analyzing the cost of producing custom-printed T-shirts for a school event. The total cost C(x) in dollars for producing x T-shirts is modeled by the expression C(x) = 16x + 121. In this expression, what does the number 16 represent in the context of the T-shirt production? Answer: The cost per T-shirt in dollars. Solution: The expression is C(x) = 16x + 121. This is a linear expression of the form mx + b, where m is the coefficient of x and b is the constant term. In context, x represents the number of T-shirts produced.
   Full step-by-step solution

   Step 1: The expression is C(x) = 16x + 121. Step 2: This is a linear expression of the form mx + b, where m is the coefficient of x and b is the constant term. Step 3: In context, x represents the number of T-shirts produced. When x increases by 1, the total cost increases by 16 dollars. Step 4: This means 16 is the additional cost for each extra T-shirt, so it represents the cost per T-shirt. Step 5: The constant term 121 represents the fixed costs (like setup, design) that do not depend on the number of T-shirts. Therefore, 16 represents the cost per T-shirt in dollars.