Quadratic Applications

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Quadratic Applications · Grade 9 · Worksheet 2

1. Ava kicks a soccer ball whose height is modeled by h(t) = -16t² + 80t + 4. Find the maximum height the ball reaches. Answer: 104 Solution: Identify the coefficients: a = -16, b = 80, c = 4 Find the time when maximum height occurs: t = -b/(2a) = -80/(2×-16) = -80/-32 = 2.5 seconds Substitute t = 2.5 into the height function: h(2.5) = -16(2.5)² + 80(2.5) + 4 Calculate (2.5)² = 6.25 Multiply: -16 × 6.25 = -100 Multiply: 80 × 2.5 = 200…
   Full step-by-step solution

   Step 1: Identify the coefficients: a = -16, b = 80, c = 4 Step 2: Find the time when maximum height occurs: t = -b/(2a) = -80/(2×-16) = -80/-32 = 2.5 seconds Step 3: Substitute t = 2.5 into the height function: h(2.5) = -16(2.5)² + 80(2.5) + 4 Step 4: Calculate (2.5)² = 6.25 Step 5: Multiply: -16 × 6.25 = -100 Step 6: Multiply: 80 × 2.5 = 200 Step 7: Combine all terms: -100 + 200 + 4 = 104 Step 8: The maximum height is 104 feet.

2. A drone is launched from a platform 25 meters high. Its height above ground is modeled by the quadratic function h(t) = -4t² + 24t + 25, where h is height in meters and t is time in seconds after launch. How many seconds after launch will the drone hit the ground? Answer: 7 Solution: Set up the equation for when the drone hits the ground: h(t) = 0 -4t² + 24t + 25 = 0 Multiply both sides by -1 to make the leading coefficient positive: 4t² - 24t - 25 = 0 Use the quadratic formula: t = [24 ± sqrt(576 + 400)] / 8 Calculate the discriminant: 576 + 400 = 976 t = [24 ± sqrt(976)] /…
   Full step-by-step solution

   Step 1: Set up the equation for when the drone hits the ground: h(t) = 0 Step 2: -4t² + 24t + 25 = 0 Step 3: Multiply both sides by -1 to make the leading coefficient positive: 4t² - 24t - 25 = 0 Step 4: Use the quadratic formula: t = [24 ± sqrt(576 + 400)] / 8 Step 5: Calculate the discriminant: 576 + 400 = 976 Step 6: t = [24 ± sqrt(976)] / 8 Step 7: Simplify sqrt(976) = sqrt(16 × 61) = 4sqrt(61) Step 8: t = [24 ± 4sqrt(61)] / 8 = [6 ± sqrt(61)] / 2 Step 9: Calculate the two solutions: t ≈ (6 - 7.81)/2 ≈ -0.91 and t ≈ (6 + 7.81)/2 ≈ 6.91 Step 10: Since time cannot be negative, the drone hits the ground at approximately 6.91 seconds, which rounds to 7 seconds. The answer is 7 seconds.

3. f(x) = 3x² - 12x + 7, f(4) = ? Answer: 7 Solution: Start with the function f(x) = 3x² - 12x + 7 Substitute x = 4 into the function: f(4) = 3(4)² - 12(4) + 7 Calculate the exponent first: (4)² = 16 Multiply: 3 × 16 = 48 and 12 × 4 = 48 Substitute back: f(4) = 48 - 48 + 7 Add and subtract from left to right: 48 - 48 = 0, then 0 + 7 = 7 The answer…
   Full step-by-step solution

   Step 1: Start with the function f(x) = 3x² - 12x + 7 Step 2: Substitute x = 4 into the function: f(4) = 3(4)² - 12(4) + 7 Step 3: Calculate the exponent first: (4)² = 16 Step 4: Multiply: 3 × 16 = 48 and 12 × 4 = 48 Step 5: Substitute back: f(4) = 48 - 48 + 7 Step 6: Add and subtract from left to right: 48 - 48 = 0, then 0 + 7 = 7 The answer is 7.

4. Sophia throws a ball whose height is modeled by h(t) = -5t² + 20t + 8. What is the maximum height reached by the ball? Answer: 28 Solution: The height function is h(t) = -5t² + 20t + 8. This is a downward-opening parabola since a = -5 < 0, so it has a maximum at the vertex.
   Full step-by-step solution

   Step 1: The height function is h(t) = -5t² + 20t + 8. This is a downward-opening parabola since a = -5 < 0, so it has a maximum at the vertex. Step 2: Find the time when maximum height occurs using t = -b/(2a) = -20/(2×(-5)) = -20/(-10) = 2 seconds. Step 3: Substitute t = 2 into the height function: h(2) = -5(2)² + 20(2) + 8 = -5(4) + 40 + 8 = -20 + 40 + 8 = 28. Step 4: The maximum height is 28 units.

5. Nova throws a ball straight upward from a platform 6 meters above the ground. The height of the ball above the ground, h(t) in meters, after t seconds is given by the quadratic function h(t) = -5t² + 16t + 6. The graph of this function is a parabola opening downward, with the vertex representing the maximum height of the ball. After how many seconds does the ball reach its maximum height, and what is that maximum height? Answer: 1.6 seconds; 18.8 meters Solution: The height function is h(t) = -5t² + 16t + 6. This is a quadratic in standard form where a = -5, b = 16, c = 6. The ball reaches its maximum height at the vertex of the parabola.
   Full step-by-step solution

   Step 1: The height function is h(t) = -5t² + 16t + 6. This is a quadratic in standard form where a = -5, b = 16, c = 6. Step 2: The ball reaches its maximum height at the vertex of the parabola. The time at the vertex is given by the axis of symmetry formula: t = -b / (2a). Step 3: Substitute the values: t = -16 / (2 * -5) = -16 / -10 = 1.6 seconds. Step 4: To find the maximum height, substitute t = 1.6 into the height function: h(1.6) = -5(1.6)² + 16(1.6) + 6 Step 5: Calculate (1.6)² = 2.56. Then -5 * 2.56 = -12.8. Step 6: 16 * 1.6 = 25.6. Step 7: Now add: h(1.6) = -12.8 + 25.6 + 6 = 12.8 + 6 = 18.8 meters. Step 8: Therefore, the ball reaches its maximum height of 18.8 meters after 1.6 seconds. Answer: 1.6 seconds; 18.8 meters.

6. A company's profit P(x) from selling x units of a product is modeled by the quadratic function P(x) = -2x² + 120x - 1000. How many units must be sold to maximize the profit? Answer: 30 Solution: The profit function is P(x) = -2x² + 120x - 1000 Since the coefficient of x² is negative (-2), the parabola opens downward, so the vertex represents the maximum point The x-coordinate of the vertex for a quadratic function ax² + bx + c is given by x = -b/(2a) Substitute a = -2 and b = 120 into…
   Full step-by-step solution

   Step 1: The profit function is P(x) = -2x² + 120x - 1000 Step 2: Since the coefficient of x² is negative (-2), the parabola opens downward, so the vertex represents the maximum point Step 3: The x-coordinate of the vertex for a quadratic function ax² + bx + c is given by x = -b/(2a) Step 4: Substitute a = -2 and b = 120 into the formula: x = -120/(2*(-2)) Step 5: Calculate the denominator: 2*(-2) = -4 Step 6: Complete the calculation: x = -120/(-4) = 30 Step 7: Therefore, 30 units must be sold to maximize profit The answer is 30.

7. Mason throws a ball whose height is modeled by h(t) = -12t² + 72t + 7. Find the maximum height. Answer: 115 Solution: The height function is h(t) = -12t² + 72t + 7. This is a downward-opening parabola since a = -12 < 0. The maximum height occurs at the vertex.
   Full step-by-step solution

   Step 1: The height function is h(t) = -12t² + 72t + 7. This is a downward-opening parabola since a = -12 < 0. Step 2: The maximum height occurs at the vertex. For a quadratic in the form at² + bt + c, the t-coordinate of the vertex is t = -b/(2a). Step 3: Calculate t = -72/(2×(-12)) = -72/(-24) = 3 seconds. Step 4: Substitute t = 3 into the height function: h(3) = -12(3)² + 72(3) + 7 Step 5: Calculate: -12(9) + 72(3) + 7 = -108 + 216 + 7 Step 6: Simplify: -108 + 216 = 108, then 108 + 7 = 115 Step 7: The maximum height is 115 feet. The answer is 115.