Interpret Expressions

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Answer Key & Explanations

Interpret Expressions · Grade 9 · Worksheet 3

1. Moana is studying a ball thrown upward. The height of the ball in meters after 3 seconds is given by the expression -5*3*3 + 30*3 + 10. Interpret the expression to explain what the quantity 10 represents in this context. Answer: 10 Solution: The expression is -5*3*3 + 30*3 + 10. This is a quadratic model for the height of the ball over time. The variable 3 is time in seconds.
   Full step-by-step solution

   Step 1: The expression is -5*3*3 + 30*3 + 10. Step 2: This is a quadratic model for the height of the ball over time. Step 3: The variable 3 is time in seconds. When 3 = 0 (at the moment of release), the first two terms become zero. Step 4: At 3 = 0, the height is exactly 10. Step 5: Therefore, 10 represents the initial height of the ball in meters at the moment it was thrown. The answer is: 10 represents the initial height of the ball in meters.

2. The total cost C (in dollars) for Oliver's school club to produce 166 custom T-shirts is given by C = 20x + 182. In this expression, what quantity does the term 182 represent? Answer: 182 Solution: The cost function is C = 20x + 182, where x is the number of T-shirts. The term 20x represents the variable cost: $20 per shirt times x shirts. The constant term 182 does not depend on x.
   Full step-by-step solution

   Step 1: The cost function is C = 20x + 182, where x is the number of T-shirts. Step 2: The term 20x represents the variable cost: $20 per shirt times x shirts. Step 3: The constant term 182 does not depend on x. It represents the fixed costs (setup, design, equipment) that are incurred even if no shirts are produced. Step 4: Therefore, 182 represents the fixed cost of $182. Answer: $182 fixed cost.

3. Liam's water tank initially contains 50 gallons of water. A pump adds water at a rate of 5 gallons per minute. The expression 50 + 5t represents the total amount of water in the tank after t minutes. What quantity does the number 5 represent in this context? Answer: 5 Solution: The expression is 50 + 5t, where t represents time in minutes. The number 50 is the initial amount of water in the tank, which does not depend on time. The term 5t represents the additional water added over time.
   Full step-by-step solution

   Step 1: The expression is 50 + 5t, where t represents time in minutes. Step 2: The number 50 is the initial amount of water in the tank, which does not depend on time. Step 3: The term 5t represents the additional water added over time. Step 4: Since t is in minutes, 5t means that for each minute, 5 gallons are added. Step 5: Therefore, the number 5 represents the rate at which water is added, which is 5 gallons per minute. The answer is 5.

4. Noah's rocket height is modeled by h(t) = -16t² + 96t + 16, where t is time in seconds. What quantity does the constant term 16 represent in this context? Answer: 16 Solution: The function h(t) = -16t² + 96t + 16 gives the height of the rocket at time t seconds. The constant term is the term that does not depend on t. Here, it is 16.
   Full step-by-step solution

   Step 1: The function h(t) = -16t² + 96t + 16 gives the height of the rocket at time t seconds. Step 2: The constant term is the term that does not depend on t. Here, it is 16. Step 3: To understand what the constant term represents, evaluate h(0): h(0) = -16(0)² + 96(0) + 16 = 16. Step 4: This means at t = 0 seconds (the moment of launch), the rocket's height is 16 units. Step 5: Therefore, the constant term 16 represents the initial height of the rocket at launch. The answer is 16.

5. The total cost C (in dollars) for Kaia's school club to produce 188 custom T-shirts is given by C = 28x + 146. In this expression, what quantity does the term 146 represent? Answer: 146 Solution: The cost function is C = 28x + 146, where x is the number of T-shirts. The term 28x represents the variable cost: $28 per shirt times x shirts. The constant term 146 does not depend on x.
   Full step-by-step solution

   Step 1: The cost function is C = 28x + 146, where x is the number of T-shirts. Step 2: The term 28x represents the variable cost: $28 per shirt times x shirts. Step 3: The constant term 146 does not depend on x. It represents the fixed costs (setup, design, equipment) that are incurred even if no shirts are produced. Step 4: Therefore, 146 represents the fixed cost of $146. Answer: $146 fixed cost.

6. Liam is designing a rectangular garden with a length that is 3 meters more than twice its width. He needs to calculate the area of the garden when the width is 4 meters. Write an algebraic expression for the area in terms of the width, then evaluate it for a width of 4 meters. Answer: 44 Solution: Let the width of the garden be \( w \) meters. The length is 3 meters more than twice the width. So, length \( l = 2w + 3 \).
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Define variables** Let the width of the garden be \( w \) meters. The length is 3 meters more than twice the width. So, length \( l = 2w + 3 \). --- **Step 2: Write the area expression** Area \( A \) of a rectangle is length × width. So: \[ A = l \times w \] Substitute \( l = 2w + 3 \): \[ A = (2w + 3) \times w \] Simplify: \[ A = 2w^2 + 3w \] So the algebraic expression for the area in terms of \( w \) is: \[ A = 2w^2 + 3w \] --- **Step 3: Evaluate for width \( w = 4 \) meters** Substitute \( w = 4 \) into \( A = 2w^2 + 3w \): First compute \( w^2 = 4^2 = 16 \). Then \( 2w^2 = 2 \times 16 = 32 \). Then \( 3w = 3 \times 4 = 12 \). Add: \( 32 + 12 = 44 \). --- **Step 4: Conclusion** The area when the width is 4 meters is 44 square meters. --- **Final answer:** 44

7. Mere is analyzing the design of a parabolic satellite dish. The cross-section of the dish can be modeled on a coordinate plane, where the vertex of the parabola is at (0, 0) and the dish opens upward. The focus of the parabola is at (0, 12). A structural support beam runs along the directrix of the parabola. What is the equation of the directrix of this parabola? Answer: y = -12 Solution: Identify the vertex and focus. The vertex is at (0, 0) and the focus is at (0, 12). Since the parabola opens upward, the directrix is a horizontal line below the vertex.
   Full step-by-step solution

   Step 1: Identify the vertex and focus. The vertex is at (0, 0) and the focus is at (0, 12). Since the parabola opens upward, the directrix is a horizontal line below the vertex. Step 2: The distance from the vertex to the focus is 12 - 0 = 12 units. This distance is called p, so p = 12. Step 3: For a vertical parabola, the directrix is a horizontal line given by y = k - p, where (h, k) is the vertex. Here, h = 0 and k = 0. Step 4: Substitute into the formula: y = 0 - 12 = -12. The equation of the directrix is y = -12.

8. A scientist is studying bacterial growth in a lab culture. The population P(t) after t hours is modeled by the exponential function P(t) = 500 * 2^(t/3). If the scientist needs to determine when the population will reach 4000 bacteria, how many hours will it take? 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: 500 * 2^(9/3) = 500 * 2^3 = 500 * 8 = 4000 The…
   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: 500 * 2^(9/3) = 500 * 2^3 = 500 * 8 = 4000 The answer is 9 hours.