Interpret Expressions

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Answer Key & Explanations

Interpret Expressions · Grade 9 · Worksheet 1

1. Aisha is analyzing the growth of a bacteria culture in her biology lab. The population P after t hours is modeled by the function P(t) = 500 * 2^(t/3). She needs to determine how long it will take for the bacteria population to reach 4000. Write an equation and solve for t. Answer: 9 Solution: Set up the equation using the given function: 500 * 2^(t/3) = 4000 Divide both sides by 500: 2^(t/3) = 8 Recognize that 8 = 2^3, so 2^(t/3) = 2^3 Since the bases are equal, set the exponents equal: t/3 = 3 Multiply both sides by 3: t = 9 Check: P(9) = 500 * 2^(9/3) = 500 * 2^3 = 500 * 8 = 4000…
   Full step-by-step solution

   Step 1: Set up the equation using the given function: 500 * 2^(t/3) = 4000 Step 2: Divide both sides by 500: 2^(t/3) = 8 Step 3: Recognize that 8 = 2^3, so 2^(t/3) = 2^3 Step 4: Since the bases are equal, set the exponents equal: t/3 = 3 Step 5: Multiply both sides by 3: t = 9 Step 6: Check: P(9) = 500 * 2^(9/3) = 500 * 2^3 = 500 * 8 = 4000 The answer is 9 hours.

2. Noah's rocket height is modeled by h(t) = -16t² + 48t + 64 where t represents time in seconds. What quantity does the constant term 64 represent in this context? Answer: the initial height of the rocket in feet Solution: The function h(t) = -16t² + 48t + 64 gives the height of the rocket at time t seconds. Step 2: The constant term is 64, which does not involve t. Step 3: When t = 0, h(0) = -16(0)² + 48(0) + 64 = 64.
   Full step-by-step solution

   Step 1: The function h(t) = -16t² + 48t + 64 gives the height of the rocket at time t seconds. Step 2: The constant term is 64, which does not involve t. Step 3: When t = 0, h(0) = -16(0)² + 48(0) + 64 = 64. Step 4: At t = 0, the rocket has not yet launched, so its height is the starting height above ground. Step 5: Therefore, the constant term 64 represents the initial height of the rocket in feet. The answer is: the initial height of the rocket in feet.

3. Olivia's company profit is modeled by P(t) = -5t² + 40t + 50, where t represents time in years since the company started. What does the constant term 50 represent in this context? Answer: 50 Solution: The profit function is P(t) = -5t² + 40t + 50, where t is time in years since the company started. The constant term is 50, which is independent of t.
   Full step-by-step solution

   Step 1: The profit function is P(t) = -5t² + 40t + 50, where t is time in years since the company started. Step 2: The constant term is 50, which is independent of t. Step 3: When t = 0 (the year the company started), P(0) = -5(0)² + 40(0) + 50 = 50. Step 4: This means the company's profit at the start (year 0) is 50 (in thousands of dollars). Step 5: Therefore, the constant term 50 represents the initial profit of the company when it first began operations. The answer is 50.

4. Liam is designing a rectangular garden with a perimeter of 60 meters. He wants to model the area A of the garden as a function of its width w. Write an algebraic expression for A(w) and determine the maximum possible area of the garden. Answer: 225 Solution: We have a rectangular garden with perimeter \( P = 60 \) meters. Let \( w \) = width (in meters), \( l \) = length (in meters). \( P = 2l + 2w \) So \( 2l + 2w = 60 \).
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** We have a rectangular garden with perimeter \( P = 60 \) meters. Let \( w \) = width (in meters), \( l \) = length (in meters). Perimeter formula: \( P = 2l + 2w \) So \( 2l + 2w = 60 \). --- **Step 2: Solve for \( l \) in terms of \( w \)** \( 2l + 2w = 60 \) Divide by 2: \( l + w = 30 \) Thus \( l = 30 - w \). --- **Step 3: Write the area \( A \) as a function of \( w \)** Area \( A = l \times w \) Substitute \( l = 30 - w \): \( A(w) = (30 - w) \times w \) \( A(w) = 30w - w^2 \). --- **Step 4: Find the maximum possible area** \( A(w) = -w^2 + 30w \) is a downward-opening parabola (since coefficient of \( w^2 \) is negative). Maximum occurs at the vertex. For a quadratic \( aw^2 + bw + c \), vertex at \( w = -b/(2a) \). Here \( a = -1 \), \( b = 30 \): \( w = -30 / (2 \times -1) = -30 / -2 = 15 \). So width \( w = 15 \) m gives maximum area. --- **Step 5: Find the maximum area** Substitute \( w = 15 \) into \( A(w) \): \( A(15) = 30(15) - (15)^2 \) \( A(15) = 450 - 225 \) \( A(15) = 225 \). --- **Step 6: Conclusion** The algebraic expression for the area is \( A(w) = 30w - w^2 \). The maximum possible area is \( 225 \) square meters. --- **Final answer:** 225

5. Noah's company profit is modeled by P(x) = 9x² - 17x + 13 where x represents hundreds of units sold. What does the constant term 13 represent in this context? Answer: 13 Solution: In the profit function P(x) = 9x² - 17x + 13, the variable x represents hundreds of units sold. The constant term 13 is independent of x, meaning it does not change based on the number of units sold.
   Full step-by-step solution

   Step 1: In the profit function P(x) = 9x² - 17x + 13, the variable x represents hundreds of units sold. Step 2: The constant term 13 is independent of x, meaning it does not change based on the number of units sold. Step 3: To find what it represents, evaluate P(0): P(0) = 9(0)² - 17(0) + 13 = 13. Step 4: This means when zero units are sold, the profit is 13 (in appropriate monetary units). Step 5: In business profit functions, constant terms typically represent fixed costs or initial investments. Since the constant is positive, it likely represents initial capital or fixed revenue sources. The answer is 13.

6. Matiu's drone is launched from a platform. Its height in meters after t seconds is given by h(t) = -4t² + 16t + 20. What quantity does the constant term 20 represent in this context? Answer: 20 Solution: The height function is h(t) = -4t² + 16t + 20, where t is time in seconds. The constant term is the part of the expression that does not depend on t.
   Full step-by-step solution

   Step 1: The height function is h(t) = -4t² + 16t + 20, where t is time in seconds. Step 2: The constant term is the part of the expression that does not depend on t. Step 3: To find its meaning, evaluate h(0): h(0) = -4(0)² + 16(0) + 20 = 0 + 0 + 20 = 20. Step 4: This shows that at t = 0 seconds, the drone's height is 20 meters. Step 5: Therefore, the constant term 20 represents the initial height of the drone above ground (the height of the launch platform). The answer is 20.

7. Mere's rectangular garden has length 12 meters and width 8 meters. The expression 2(12 + 8) represents what quantity? Answer: 40 Solution: The expression is 2(12 + 8), where 12 represents the length and 8 represents the width First, add the length and width: 12 + 8 = 20 Then multiply by 2: 2 × 20 = 40 This calculation gives us the perimeter of the rectangle The perimeter represents the total distance around Mere's garden The answer…
   Full step-by-step solution

   Step 1: The expression is 2(12 + 8), where 12 represents the length and 8 represents the width Step 2: First, add the length and width: 12 + 8 = 20 Step 3: Then multiply by 2: 2 × 20 = 40 Step 4: This calculation gives us the perimeter of the rectangle Step 5: The perimeter represents the total distance around Mere's garden The answer is 40 meters.