Functions in Context

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Answer Key & Explanations

Functions in Context · Grade 9 · Worksheet 3

1. The function h(t) = -5t² + 45t models the height in meters of a ball thrown by Aroha after t seconds. What does h(3) represent and what is its value? Answer: 90 Solution: The function h(t) = -5t² + 45t models the height of Aroha's ball after t seconds. h(3) represents the height of the ball after 3 seconds.
   Full step-by-step solution

   Step 1: The function h(t) = -5t² + 45t models the height of Aroha's ball after t seconds. Step 2: h(3) represents the height of the ball after 3 seconds. Step 3: Substitute t = 3 into the function: h(3) = -5(3)² + 45(3) Step 4: Calculate the exponent: (3)² = 9 Step 5: Multiply: -5 × 9 = -45 Step 6: Multiply: 45 × 3 = 135 Step 7: Add the results: -45 + 135 = 90 Step 8: The height after 3 seconds is 90 meters.

2. Mere is tracking the height of a plant in her science experiment. The height h(t) in centimeters after t days is modeled by the function h(t) = 2.5t + 15. What does h(10) represent in this context, and what is its value? Answer: h(10) represents the height of the plant after 10 days, which is 40 centimeters. Solution: Identify the meaning of the function. h(t) = 2.5t + 15 models the height of the plant in centimeters after t days. The number 15 is the initial height (when t = 0), and 2.5 is the growth rate in centimeters per day.
   Full step-by-step solution

   Step 1: Identify the meaning of the function. h(t) = 2.5t + 15 models the height of the plant in centimeters after t days. The number 15 is the initial height (when t = 0), and 2.5 is the growth rate in centimeters per day. Step 2: Interpret h(10). It means the height of the plant after 10 days. Step 3: Calculate h(10). Substitute t = 10 into the equation: h(10) = 2.5(10) + 15 = 25 + 15 = 40. Step 4: State the final answer. h(10) = 40, meaning the plant is 40 centimeters tall after 10 days. The answer is: h(10) represents the height of the plant after 10 days, which is 40 centimeters.

3. A drone is flying over a park following a parabolic path modeled by the function h(t) = -2t² + 12t + 5, where h represents the drone's height in meters and t represents time in seconds. At what time does the drone reach its maximum height, and what is that maximum height? Answer: 3 seconds, 23 meters Solution: h(t) = -2t² + 12t + 5 This is a quadratic function in the form h(t) = at² + bt + c, where: a = -2, b = 12, c = 5 Since a < 0, the parabola opens downward, so it has a maximum at its vertex.
   Full step-by-step solution

   Let's solve this step by step. We are given the height function: h(t) = -2t² + 12t + 5 --- **Step 1: Identify the type of function** This is a quadratic function in the form h(t) = at² + bt + c, where: a = -2, b = 12, c = 5 Since a < 0, the parabola opens downward, so it has a maximum at its vertex. --- **Step 2: Find the time at maximum height** For a quadratic function at² + bt + c, the vertex occurs at: t = -b / (2a) Substitute b = 12 and a = -2: t = -12 / (2 * -2) t = -12 / (-4) t = 3 So the drone reaches its maximum height at t = 3 seconds. --- **Step 3: Find the maximum height** Substitute t = 3 into h(t): h(3) = -2*(3)² + 12*(3) + 5 h(3) = -2*9 + 36 + 5 h(3) = -18 + 36 + 5 h(3) = 18 + 5 h(3) = 23 So the maximum height is 23 meters. --- **Final Answer:** Time = 3 seconds, Maximum height = 23 meters

4. A company's profit is modeled by the quadratic function P(x) = -2x² + 120x - 1000, where x represents the number of units sold (in hundreds) and P(x) is the profit in thousands of dollars. At what number of units sold does the company reach its maximum profit? Answer: 3000 Solution: P(x) = -2x² + 120x - 1000 Here, x is in hundreds of units, and P(x) is in thousands of dollars.
   Full step-by-step solution

   We are given the profit function: P(x) = -2x² + 120x - 1000 Here, x is in hundreds of units, and P(x) is in thousands of dollars. --- **Step 1: Identify the type of function** This is a quadratic function of the form ax² + bx + c, with: a = -2 b = 120 c = -1000 Since a < 0, the parabola opens downward, so the vertex gives the maximum profit. --- **Step 2: Vertex formula** For a quadratic function, the x-coordinate of the vertex is: x = -b / (2a) Substitute b = 120 and a = -2: x = -120 / (2 * -2) x = -120 / (-4) x = 120 / 4 x = 30 --- **Step 3: Interpret x** x is in hundreds of units, so: Number of units = x * 100 = 30 * 100 = 3000 units. --- **Step 4: Conclusion** The company reaches maximum profit when 3000 units are sold. --- **Final answer:** 3000

5. Liam is analyzing the profit function for his small business selling handmade candles. The profit P(x) in dollars is modeled by the quadratic function P(x) = -2x² + 80x - 600, where x represents the number of candles sold. How many candles must Liam sell to maximize his profit? Answer: 20 Solution: To find the number of candles Liam must sell to maximize profit, we use the profit function: P(x) = -2x² + 80x - 600 This is a quadratic function in the form ax² + bx + c, where: a = -2 b = 80 c = -600 Since the coefficient of x² (a = -2) is negative, the parabola opens downward.
   Full step-by-step solution

   To find the number of candles Liam must sell to maximize profit, we use the profit function: P(x) = -2x² + 80x - 600 This is a quadratic function in the form ax² + bx + c, where: a = -2 b = 80 c = -600 Since the coefficient of x² (a = -2) is negative, the parabola opens downward. This means the function has a maximum value at its vertex. For any quadratic function ax² + bx + c, the x-coordinate of the vertex (which gives the maximum or minimum point) is found using the formula: x = -b / (2a) Let's substitute our values for a and b: x = -80 / (2 × -2) First, calculate the denominator: 2 × -2 = -4 So, x = -80 / (-4) Dividing a negative by a negative gives a positive: x = 80 / 4 x = 20 Therefore, Liam must sell 20 candles to maximize his profit. Verification: At x = 20, the profit would be: P(20) = -2(20)² + 80(20) - 600 = -2(400) + 1600 - 600 = -800 + 1600 - 600 = 800 - 600 = 200 dollars This confirms that selling 20 candles gives a profit of $200, and since the parabola opens downward, this is indeed the maximum profit point. ANSWER: 20

6. Emma's car depreciates according to V(t) = 25000(0.87)^t where t is years. What does V(3) represent? Answer: The car's value after 3 years Solution: Step 1: The function V(t) = 25000(0.87)^t models car value over time Step 2: V(3) means we're evaluating the function when t = 3 Step 3: In this context, t represents years since purchase Step 4: Therefore, V(3) represents the car's monetary value after 3 years of ownership Step 5: The answer…
   Full step-by-step solution

   Step 1: The function V(t) = 25000(0.87)^t models car value over time Step 2: V(3) means we're evaluating the function when t = 3 Step 3: In this context, t represents years since purchase Step 4: Therefore, V(3) represents the car's monetary value after 3 years of ownership Step 5: The answer describes what this specific function evaluation means in the real-world scenario

7. Sophia is studying the temperature of a chemical solution in her lab. The temperature T(t) in degrees Celsius after t minutes is modeled by the quadratic function T(t) = -3t² + 24t + 10. What is the maximum temperature the solution reaches, and after how many minutes does it occur? Answer: 58 degrees Celsius at 4 minutes Solution: The function T(t) = -3t² + 24t + 10 is a quadratic with a = -3, b = 24, and c = 10. Since a is negative, the parabola opens downward, so the vertex gives the maximum temperature.
   Full step-by-step solution

   Step 1: The function T(t) = -3t² + 24t + 10 is a quadratic with a = -3, b = 24, and c = 10. Since a is negative, the parabola opens downward, so the vertex gives the maximum temperature. Step 2: Find the time t when the maximum occurs using the vertex formula t = -b/(2a). t = -24/(2 * -3) = -24/(-6) = 4 minutes. Step 3: Find the maximum temperature by substituting t = 4 into the function. T(4) = -3(4)² + 24(4) + 10 = -3(16) + 96 + 10 = -48 + 96 + 10 = 58 degrees Celsius. The maximum temperature is 58 degrees Celsius, reached after 4 minutes.

8. Noah is tracking the height of a bouncing ball. The height h(t) in meters after t bounces is modeled by the function h(t) = 16 × (1/2)^t. Interpret what h(4) represents in this context, and calculate its value. Answer: h(4) = 1 meter; it represents the height of the ball after 4 bounces. Solution: Identify the function: h(t) = 16 × (1/2)^t. We want to find h(4), which means t = 4 bounces. Substitute t = 4: h(4) = 16 × (1/2)^4.
   Full step-by-step solution

   Step 1: Identify the function: h(t) = 16 × (1/2)^t. Step 2: We want to find h(4), which means t = 4 bounces. Step 3: Substitute t = 4: h(4) = 16 × (1/2)^4. Step 4: Calculate (1/2)^4 = 1/16. Step 5: Multiply: 16 × 1/16 = 1. Step 6: Interpretation: h(4) = 1 meter means that after 4 bounces, the ball reaches a height of 1 meter. The answer is h(4) = 1 meter; it represents the height of the ball after 4 bounces.