Functions in Context

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Answer Key & Explanations

Functions in Context · Grade 9 · Worksheet 2

1. Hana is tracking the height of a model rocket launched from the ground. The rocket's height in meters, h(t), after t seconds is modeled by the function h(t) = -4t² + 32t. Interpret the meaning of h(4) in the context of this problem and calculate its value. Answer: 64 meters, the rocket's height 4 seconds after launch Solution: Identify the meaning of h(4). The function h(t) gives the height in meters at time t seconds. So h(4) represents the height of the rocket 4 seconds after launch.
   Full step-by-step solution

   Step 1: Identify the meaning of h(4). The function h(t) gives the height in meters at time t seconds. So h(4) represents the height of the rocket 4 seconds after launch. Step 2: Substitute t = 4 into the function: h(4) = -4(4)² + 32(4) Step 3: Calculate 4² = 16, so h(4) = -4(16) + 128 Step 4: Multiply: -4 × 16 = -64, so h(4) = -64 + 128 Step 5: Add: -64 + 128 = 64 The answer is 64 meters. This means that 4 seconds after launch, the rocket is 64 meters above the ground.

2. Liam is designing a parabolic arch for a bridge. The arch's height above the ground is modeled by the function h(x) = -0.02x² + 1.2x, where x is the horizontal distance in meters from the left support and h(x) is the height in meters. What is the maximum height of the arch above the ground? Answer: 18 Solution: h(x) = -0.02x² + 1.2x This is a quadratic function in the form ax² + bx + c, where: a = -0.02 b = 1.2 c = 0 Since a < 0, the parabola opens downward, so the vertex gives the maximum height.
   Full step-by-step solution

   We are given the function for the arch height: h(x) = -0.02x² + 1.2x This is a quadratic function in the form ax² + bx + c, where: a = -0.02 b = 1.2 c = 0 Since a < 0, the parabola opens downward, so the vertex gives the maximum height. The x-coordinate of the vertex is: x = -b / (2a) Substitute b and a: x = -1.2 / (2 * -0.02) x = -1.2 / (-0.04) x = 1.2 / 0.04 x = 120 / 4 x = 30 So the maximum height occurs at x = 30 meters. Now substitute x = 30 into h(x): h(30) = -0.02 * (30)² + 1.2 * 30 First compute (30)² = 900. Then -0.02 * 900 = -18. Then 1.2 * 30 = 36. So h(30) = -18 + 36 = 18. Therefore, the maximum height of the arch is 18 meters.

3. Mere is analyzing the graph of a quadratic function that models the height (in meters) of a soccer ball kicked from the ground. The function is h(t) = -5t^2 + 20t, where t is time in seconds. The graph is a parabola opening downward, crossing the t-axis at t = 0 and t = 4. What does the vertex of this parabola represent in the context of the soccer ball's flight? Answer: The maximum height of the soccer ball is 20 meters, reached at 2 seconds. Solution: The quadratic function h(t) = -5t^2 + 20t models the height of the soccer ball over time. The graph is a parabola opening downward because the coefficient of t^2 is negative (-5).
   Full step-by-step solution

   Step 1: The quadratic function h(t) = -5t^2 + 20t models the height of the soccer ball over time. The graph is a parabola opening downward because the coefficient of t^2 is negative (-5). Step 2: The vertex of a parabola given by h(t) = at^2 + bt + c occurs at t = -b/(2a). Here, a = -5 and b = 20, so t = -20/(2 * -5) = -20/(-10) = 2 seconds. Step 3: Substitute t = 2 into the function to find the height: h(2) = -5(2)^2 + 20(2) = -5(4) + 40 = -20 + 40 = 20 meters. Step 4: The vertex (2, 20) is the highest point on the graph. In context, this means the soccer ball reaches its maximum height of 20 meters at 2 seconds after being kicked. The answer is: The maximum height of the soccer ball is 20 meters, reached at 2 seconds.

4. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (6,8). A circle is inscribed inside this triangle such that it is tangent to all three sides. What is the radius of this inscribed circle? Answer: 2 Solution: The triangle has vertices at (0,0), (6,0), and (6,8). - Side along x-axis from (0,0) to (6,0) is length 6. - Vertical side from (6,0) to (6,8) is length 8.
   Full step-by-step solution

   Step 1: Understand the triangle and circle The triangle has vertices at (0,0), (6,0), and (6,8). - Side along x-axis from (0,0) to (6,0) is length 6. - Vertical side from (6,0) to (6,8) is length 8. - The third side from (6,8) to (0,0) is the hypotenuse. Step 2: Find the hypotenuse length Using the distance formula between (0,0) and (6,8): Hypotenuse = sqrt((6-0)^2 + (8-0)^2) = sqrt(36 + 64) = sqrt(100) = 10. So the triangle sides are: a = 8 (vertical leg), b = 6 (horizontal leg), c = 10 (hypotenuse). Step 3: Formula for inradius of a right triangle For any right triangle with legs a and b, hypotenuse c, the inradius r is: r = (a + b - c)/2. Step 4: Apply the formula r = (8 + 6 - 10)/2 = (4)/2 = 2. Step 5: Explanation of the formula The inradius formula for a right triangle comes from the fact that the area Δ = (1/2)*a*b, and also Δ = r*s, where s is the semiperimeter s = (a+b+c)/2. Equating: (1/2)*a*b = r*(a+b+c)/2 → a*b = r*(a+b+c) → r = (a*b)/(a+b+c). But for a right triangle, a+b-c = 2r is another known property (can be derived from equal tangents to the two legs: a = r + x, b = r + y, c = x + y, then a+b = 2r + (x+y) = 2r + c → a+b-c = 2r). Let's check: Using r = (a*b)/(a+b+c) = (8*6)/(8+6+10) = 48/24 = 2. Both formulas give r = 2. Step 6: Conclusion The radius of the inscribed circle is 2.

5. Mason is analyzing the flight path of a toy rocket launched from a platform. The height of the rocket in meters above the ground, h(t), is modeled by the quadratic function h(t) = -4.9t^2 + 98t + 10, where t is the time in seconds after launch. The graph of this function is a parabola opening downward. Interpret the meaning of the coordinates of the vertex of this parabola in the context of the rocket's flight. What is the maximum height reached by the rocket? (Round your answer to the nearest whole meter.) Answer: 500 Solution: Identify the coefficients from the quadratic function h(t) = -4.9t^2 + 98t + 10. Here a = -4.9, b = 98, and c = 10. The t-coordinate of the vertex is given by t = -b/(2a).
   Full step-by-step solution

   Step 1: Identify the coefficients from the quadratic function h(t) = -4.9t^2 + 98t + 10. Here a = -4.9, b = 98, and c = 10. Step 2: The t-coordinate of the vertex is given by t = -b/(2a). Substitute the values: t = -98 / (2 * -4.9) = -98 / -9.8 = 10 seconds. Step 3: The h-coordinate of the vertex (the maximum height) is found by substituting t = 10 into the function: h(10) = -4.9(10)^2 + 98(10) + 10 = -4.9(100) + 980 + 10 = -490 + 980 + 10 = 500. Step 4: The vertex is at (10, 500). In context, the t-coordinate 10 means the rocket reaches its maximum height at 10 seconds after launch. The h-coordinate 500 means the maximum height is 500 meters above the ground. The maximum height reached by the rocket is 500 meters.

6. The function P(t) = 2500(1.07)^t models Olivia's investment growth. What does P(9) represent? Answer: The value of Olivia's investment after 9 years Solution: The function P(t) = 2500(1.07)^t models investment growth over time The initial investment is $2500 The growth rate is 7% per year (from 1.07) P(9) means we substitute t = 9 into the function This calculates the investment value after 9 years of growth Therefore, P(9) represents the value of…
   Full step-by-step solution

   Step 1: The function P(t) = 2500(1.07)^t models investment growth over time Step 2: The initial investment is $2500 Step 3: The growth rate is 7% per year (from 1.07) Step 4: P(9) means we substitute t = 9 into the function Step 5: This calculates the investment value after 9 years of growth Step 6: Therefore, P(9) represents the value of Olivia's investment after 9 years