Functions in Context

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Answer Key & Explanations

Functions in Context · Grade 9 · Worksheet 2

1. Liam is designing a parabolic arch for a garden bridge. The arch follows the function h(x) = -0.5x² + 4x, where h(x) represents the height in meters above the ground at a horizontal distance x meters from the left support. What is the maximum height of the arch above the ground? Answer: 8 meters Solution: h(x) = -0.5x² + 4x The coefficient of x² is -0.5, which is negative. That means the parabola opens downward, so the vertex of the parabola gives the maximum height.
   Full step-by-step solution

   Let's find the maximum height of the arch step by step. The height function is: h(x) = -0.5x² + 4x --- **Step 1: Recognize the shape of the parabola** The coefficient of x² is -0.5, which is negative. That means the parabola opens downward, so the vertex of the parabola gives the maximum height. --- **Step 2: Find the x-coordinate of the vertex** For a quadratic function in the form ax² + bx + c, the x-coordinate of the vertex is given by: x = -b / (2a) Here, a = -0.5, b = 4. So: x = -4 / (2 * -0.5) x = -4 / (-1) x = 4 So the maximum height occurs at x = 4 meters from the left support. --- **Step 3: Calculate the height at x = 4** Substitute x = 4 into h(x): h(4) = -0.5 * (4)² + 4 * 4 h(4) = -0.5 * 16 + 16 h(4) = -8 + 16 h(4) = 8 --- **Step 4: State the final answer** The maximum height of the arch above the ground is 8 meters. --- **Final answer:** 8 meters

2. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (6,8). A circle is circumscribed around this triangle such that all three vertices lie on the circle's circumference. What is the area of this circumscribed circle? (Use π = 3.14) Answer: 78.5 Solution: We have a right triangle with vertices at (0,0), (6,0), and (6,8). The hypotenuse is between (0,0) and (6,8). A circle is circumscribed around the triangle, meaning all three vertices lie on the circle.
   Full step-by-step solution

   Step 1: Understand the problem We have a right triangle with vertices at (0,0), (6,0), and (6,8). The hypotenuse is between (0,0) and (6,8). A circle is circumscribed around the triangle, meaning all three vertices lie on the circle. Step 2: Recall a key geometry fact For a right triangle, the hypotenuse is the diameter of the circumscribed circle. So, the length of the hypotenuse = diameter of the circle. Step 3: Find the length of the hypotenuse The hypotenuse is between (0,0) and (6,8). Distance formula: d = sqrt((6-0)^2 + (8-0)^2) d = sqrt(36 + 64) d = sqrt(100) d = 10 So, the hypotenuse length is 10. Step 4: Relate to the circle Since the hypotenuse is the diameter of the circle: Diameter D = 10 Radius r = D/2 = 10/2 = 5 Step 5: Find the area of the circle Area = π * r^2 Area = 3.14 * (5)^2 Area = 3.14 * 25 Area = 78.5 Step 6: Final answer The area of the circumscribed circle is 78.5.

3. Isabella is a civil engineer designing a suspension bridge. The main cable's height above the bridge deck, in meters, is modeled by the quadratic function h(x) = 0.002x² - 0.48x + 36, where x is the horizontal distance in meters from the left tower. Interpret the meaning of h(120) in the context of this problem, and then calculate and state the value of h(120). Answer: h(120) = 7.2 meters; this represents the height of the main cable above the bridge deck at a horizontal distance of 120 meters from the left tower. Solution: Identify the meaning of the variables. x is the horizontal distance in meters from the left tower. Step 2: Substitute x = 120 into the function: h(120) = 0.002(120)² - 0.48(120) + 36.
   Full step-by-step solution

   Step 1: Identify the meaning of the variables. x is the horizontal distance in meters from the left tower. h(x) is the height of the cable above the bridge deck in meters. Step 2: Substitute x = 120 into the function: h(120) = 0.002(120)² - 0.48(120) + 36. Step 3: Calculate (120)² = 14,400. Step 4: Multiply: 0.002 × 14,400 = 28.8. Step 5: Multiply: -0.48 × 120 = -57.6. Step 6: Add the terms: 28.8 - 57.6 + 36 = 7.2. Step 7: Interpret the result: At a horizontal distance of 120 meters from the left tower, the main cable is 7.2 meters above the bridge deck. The answer is h(120) = 7.2 meters.

4. Liam is designing a skateboard ramp with a parabolic cross-section. The height of the ramp (in feet) is modeled by the function h(x) = -0.2x² + 1.6x, where x is the horizontal distance from the starting edge. What is the maximum height the ramp reaches, and at what horizontal distance does this occur? Answer: 3.2 feet at 4 feet Solution: We are given the function for the height of the ramp: h(x) = -0.2x² + 1.6x This is a quadratic function in the form ax² + bx + c, where: a = -0.2 b = 1.6 c = 0 Since a is negative, the parabola opens downward, so the vertex gives the maximum height.
   Full step-by-step solution

   We are given the function for the height of the ramp: h(x) = -0.2x² + 1.6x This is a quadratic function in the form ax² + bx + c, where: a = -0.2 b = 1.6 c = 0 Since a is negative, the parabola opens downward, so the vertex gives the maximum height. --- **Step 1: Find the x-coordinate of the vertex** For a quadratic function ax² + bx + c, the x-coordinate of the vertex is: x = -b / (2a) Substitute b = 1.6 and a = -0.2: x = -1.6 / (2 * -0.2) x = -1.6 / (-0.4) x = 4 So the maximum height occurs at x = 4 feet from the starting edge. --- **Step 2: Find the maximum height** Substitute x = 4 into h(x): h(4) = -0.2*(4)² + 1.6*(4) h(4) = -0.2*16 + 6.4 h(4) = -3.2 + 6.4 h(4) = 3.2 So the maximum height is 3.2 feet. --- **Final Answer:** Maximum height = 3.2 feet at horizontal distance = 4 feet.

5. Kaia is monitoring the water level in a reservoir. The depth of water, in meters, after t days is modeled by the function D(t) = -0.25t^2 + 5t + 7. Interpret the meaning of D(11) in the context of this situation and calculate its value. Answer: 31.75 meters Solution: D(t) gives the depth of water in meters after t days. To find D(11), substitute t = 11 into the function: D(11) = -0.25(11)^2 + 5(11) + 7 Calculate 11^2 = 121 Multiply -0.25 by 121: -0.25 × 121 = -30.25 Multiply 5 by 11: 5 × 11 = 55 Add the terms: -30.25 + 55 + 7 = 31.75 The answer is 31.75 meters.
   Full step-by-step solution

   Step 1: Understand the function. D(t) gives the depth of water in meters after t days. Step 2: To find D(11), substitute t = 11 into the function: D(11) = -0.25(11)^2 + 5(11) + 7 Step 3: Calculate 11^2 = 121 Step 4: Multiply -0.25 by 121: -0.25 × 121 = -30.25 Step 5: Multiply 5 by 11: 5 × 11 = 55 Step 6: Add the terms: -30.25 + 55 + 7 = 31.75 The answer is 31.75 meters. This means that after 11 days, the depth of water in the reservoir is 31.75 meters.

6. f(x) = 2x³ - 7x + 5, find f(3) = ? Answer: 38 Solution: Start with the function f(x) = 2x³ - 7x + 5 Substitute x = 3 into the function: f(3) = 2(3)³ - 7(3) + 5 Calculate the exponent first: (3)³ = 27 Multiply: 2 × 27 = 54 and 7 × 3 = 21 Substitute back: f(3) = 54 - 21 + 5 Perform subtraction and addition from left to right: 54 - 21 = 33, then 33 + 5…
   Full step-by-step solution

   Step 1: Start with the function f(x) = 2x³ - 7x + 5 Step 2: Substitute x = 3 into the function: f(3) = 2(3)³ - 7(3) + 5 Step 3: Calculate the exponent first: (3)³ = 27 Step 4: Multiply: 2 × 27 = 54 and 7 × 3 = 21 Step 5: Substitute back: f(3) = 54 - 21 + 5 Step 6: Perform subtraction and addition from left to right: 54 - 21 = 33, then 33 + 5 = 38 The answer is 38.

7. Mason is tracking the height of a sunflower he planted. The height of the sunflower, h(d), in centimeters, after d days since it sprouted is modeled by the function h(d) = 2.7d + 12. What does h(17) represent in this context, and what is its value? Answer: The height of the sunflower after 17 days is 57.9 centimeters. Solution: Identify the variables. d represents the number of days since the sunflower sprouted. To find h(17), substitute d = 17 into the function: h(17) = 2.7(17) + 12.
   Full step-by-step solution

   Step 1: Identify the variables. d represents the number of days since the sunflower sprouted. h(d) represents the height in centimeters after d days. Step 2: To find h(17), substitute d = 17 into the function: h(17) = 2.7(17) + 12. Step 3: Multiply 2.7 by 17: 2.7 * 17 = 45.9. Step 4: Add 12: 45.9 + 12 = 57.9. Step 5: Interpret: h(17) = 57.9 means that after 17 days, the sunflower is 57.9 centimeters tall. The answer is: The height of the sunflower after 17 days is 57.9 centimeters.

8. Sophia's smartphone battery percentage is modeled by B(t) = 85 - 12t, where t is hours of use. What does B(3) represent? Answer: 49 Solution: The function B(t) = 85 - 12t models Sophia's battery percentage after t hours of use B(3) represents the battery percentage after 3 hours of use Substitute t = 3 into the function: B(3) = 85 - 12(3) Multiply: 12 × 3 = 36 Subtract: 85 - 36 = 49 This means after 3 hours of use, Sophia's phone…
   Full step-by-step solution

   Step 1: The function B(t) = 85 - 12t models Sophia's battery percentage after t hours of use Step 2: B(3) represents the battery percentage after 3 hours of use Step 3: Substitute t = 3 into the function: B(3) = 85 - 12(3) Step 4: Multiply: 12 × 3 = 36 Step 5: Subtract: 85 - 36 = 49 Step 6: This means after 3 hours of use, Sophia's phone battery will be at 49% The answer is 49.