Quadratic Modeling

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Answer Key & Explanations

Quadratic Modeling · Grade 10 · Worksheet 1

1. A company's profit from selling x units of a product is modeled by the quadratic function P(x) = -2x² + 120x - 1000. How many units must the company sell to maximize its profit, and what is the maximum profit? Answer: 30 units, $800 Solution: To find the number of units that maximizes profit and the maximum profit, we use the quadratic function P(x) = -2x² + 120x - 1000. Identify the type of function.
   Full step-by-step solution

   To find the number of units that maximizes profit and the maximum profit, we use the quadratic function P(x) = -2x² + 120x - 1000. Step 1: Identify the type of function. This is a quadratic function of the form ax² + bx + c, where a = -2, b = 120, and c = -1000. Since the coefficient a = -2 is negative, the parabola opens downwards. This means the function has a maximum value at its vertex. Step 2: Find the x-coordinate of the vertex. For a quadratic function, the x-coordinate of the vertex is given by the formula: x = -b / (2a) Substitute the values of a and b: x = -120 / (2 * -2) x = -120 / -4 x = 30 So, the company must sell 30 units to maximize its profit. Step 3: Find the maximum profit. The maximum profit is the value of the function P(x) at x = 30. Substitute x = 30 into the profit function: P(30) = -2*(30)² + 120*(30) - 1000 First, calculate (30)² = 900. So, -2 * 900 = -1800. Next, 120 * 30 = 3600. Now, put it all together: P(30) = -1800 + 3600 - 1000 P(30) = ( -1800 + 3600 ) - 1000 P(30) = 1800 - 1000 P(30) = 800 Therefore, the maximum profit is $800. Final Answer: The company must sell 30 units to achieve a maximum profit of $800.

2. A company's profit P(x) = -2x² + 40x - 128, where x is units sold (in hundreds). Find the number of units that maximizes profit. Answer: 1000 Solution: The profit function is P(x) = -2x² + 40x - 128, which is a quadratic function in standard form ax² + bx + c. Since a = -2 is negative, the parabola opens downward, and the vertex represents the maximum point.
   Full step-by-step solution

   Step 1: The profit function is P(x) = -2x² + 40x - 128, which is a quadratic function in standard form ax² + bx + c. Step 2: Since a = -2 is negative, the parabola opens downward, and the vertex represents the maximum point. Step 3: The x-coordinate of the vertex is given by x = -b/(2a). Step 4: Substitute b = 40 and a = -2: x = -40/(2×(-2)) = -40/(-4) = 10. Step 5: Since x represents units sold in hundreds, the actual number of units is 10 × 100 = 1000. The number of units that maximizes profit is 1000.

3. Hana launches a model rocket from the ground. The height h(t) in meters after t seconds is modeled by a quadratic function. The rocket reaches a maximum height of 144 meters at 6 seconds. Find the height of the rocket after 4 seconds. Answer: 128 Solution: The vertex form is h(t) = a(t - 6)^2 + 144. At t = 0, h(0) = 0, so 0 = a(0 - 6)^2 + 144. Simplify: 0 = a(36) + 144, so 36a = -144, thus a = -4.
   Full step-by-step solution

   Step 1: The vertex form is h(t) = a(t - 6)^2 + 144. Step 2: At t = 0, h(0) = 0, so 0 = a(0 - 6)^2 + 144. Step 3: Simplify: 0 = a(36) + 144, so 36a = -144, thus a = -4. Step 4: The model is h(t) = -4(t - 6)^2 + 144. Step 5: At t = 4: h(4) = -4(4 - 6)^2 + 144 = -4(-2)^2 + 144 = -4(4) + 144 = -16 + 144 = 128. The height after 4 seconds is 128 meters.

4. Mere is a farmer who grows organic pumpkins. She notices that the number of pumpkins she can sell each week, n, depends on the price per pumpkin, p dollars, according to the relationship n = 3200 - 80p. The cost to grow and harvest each pumpkin is $6, and she has fixed weekly costs of $4000. Determine the price Mere should charge per pumpkin to maximize her weekly profit. Answer: 23 Solution: Write the revenue function. Revenue = price × quantity = p × n = p(3200 - 80p) = 3200p - 80p². Write the variable cost.
   Full step-by-step solution

   Step 1: Write the revenue function. Revenue = price × quantity = p × n = p(3200 - 80p) = 3200p - 80p². Step 2: Write the variable cost. Variable cost = cost per pumpkin × number of pumpkins = 6n = 6(3200 - 80p) = 19200 - 480p. Step 3: Write the total cost function. Total cost = fixed costs + variable costs = 4000 + (19200 - 480p) = 23200 - 480p. Step 4: Write the profit function. Profit = Revenue - Total Cost = (3200p - 80p²) - (23200 - 480p) = 3200p - 80p² - 23200 + 480p = -80p² + 3680p - 23200. Step 5: This is a quadratic in the form ap² + bp + c, where a = -80, b = 3680. Since a is negative, the parabola opens downward, and the vertex gives the maximum profit. The price at the vertex is p = -b/(2a). Step 6: Calculate p = -3680 / (2 × -80) = -3680 / -160 = 23. Step 7: Therefore, Mere should charge $23 per pumpkin to maximize her weekly profit. The answer is 23.

5. Mason launches a model rocket from a platform 2 meters above the ground. The rocket's initial upward velocity is 42 m/s. The height h (in meters) after t seconds is modeled by h(t) = -5t² + 42t + 2. Find the maximum height reached by the rocket. Answer: 90.2 Solution: Identify the coefficients: a = -5, b = 42, c = 2. The time at the vertex is t = -b/(2a) = -42/(2*(-5)) = -42/(-10) = 4.2 seconds. Substitute t = 4.2 into the height equation: h(4.2) = -5(4.2)² + 42(4.2) + 2.
   Full step-by-step solution

   Step 1: Identify the coefficients: a = -5, b = 42, c = 2. Step 2: The time at the vertex is t = -b/(2a) = -42/(2*(-5)) = -42/(-10) = 4.2 seconds. Step 3: Substitute t = 4.2 into the height equation: h(4.2) = -5(4.2)² + 42(4.2) + 2. Step 4: Calculate (4.2)² = 17.64. Then -5(17.64) = -88.2. Step 5: Calculate 42(4.2) = 176.4. Step 6: Add: h(4.2) = -88.2 + 176.4 + 2 = 90.2. The maximum height reached by the rocket is 90.2 meters.

6. Mason throws a ball upward from a height of 2 meters with an initial velocity of 27 m/s. The height h (in meters) after t seconds is given by h(t) = -5t² + 27t + 2. Find the maximum height reached by the ball. Answer: 38.45 Solution: Identify the coefficients: a = -5, b = 27, c = 2. The time at the vertex is t = -b/(2a) = -27/(2 * -5) = -27/(-10) = 2.7 seconds. Substitute t = 2.7 into the height equation: h(2.7) = -5(2.7)² + 27(2.7) + 2.
   Full step-by-step solution

   Step 1: Identify the coefficients: a = -5, b = 27, c = 2. Step 2: The time at the vertex is t = -b/(2a) = -27/(2 * -5) = -27/(-10) = 2.7 seconds. Step 3: Substitute t = 2.7 into the height equation: h(2.7) = -5(2.7)² + 27(2.7) + 2. Step 4: Calculate (2.7)² = 7.29. Step 5: -5(7.29) = -36.45. Step 6: 27(2.7) = 72.9. Step 7: h(2.7) = -36.45 + 72.9 + 2 = 38.45. The maximum height reached by the ball is 38.45 meters.

7. Mere is designing a parabolic water fountain for a public park. The water jet follows a parabolic path, with the nozzle located at ground level at the origin (0, 0). The water reaches its maximum height of 12 meters at a horizontal distance of 10 meters from the nozzle. The water then lands back on the ground at a point (20, 0). On a coordinate grid, each unit represents 1 meter. What is the height of the water at a horizontal distance of 15 meters from the nozzle? Answer: 9 Solution: Identify the vertex form of a parabola: y = a(x - h)^2 + k, where (h, k) is the vertex. Here, the vertex is at (10, 12), so h = 10 and k = 12. The equation becomes y = a(x - 10)^2 + 12.
   Full step-by-step solution

   Step 1: Identify the vertex form of a parabola: y = a(x - h)^2 + k, where (h, k) is the vertex. Here, the vertex is at (10, 12), so h = 10 and k = 12. The equation becomes y = a(x - 10)^2 + 12. Step 2: Use the point (0, 0) to find 'a'. Substitute x = 0 and y = 0: 0 = a(0 - 10)^2 + 12 0 = a(100) + 12 -12 = 100a a = -12 / 100 = -3/25 Step 3: The equation of the parabola is y = (-3/25)(x - 10)^2 + 12. Step 4: Find the height at x = 15. Substitute x = 15: y = (-3/25)(15 - 10)^2 + 12 y = (-3/25)(5)^2 + 12 y = (-3/25)(25) + 12 y = -3 + 12 y = 9 Step 5: The height of the water at 15 meters from the nozzle is 9 meters.

8. The profit P (in dollars) from selling x units of a product is modeled by P(x) = -2x² + 120x - 1000. Find the number of units that maximizes 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 gives the maximum profit Use the vertex formula x = -b/(2a) where a = -2 and b = 120 x = -120/(2 × -2) = -120/(-4) = 30 Therefore, selling 30 units maximizes…
   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 gives the maximum profit Step 3: Use the vertex formula x = -b/(2a) where a = -2 and b = 120 Step 4: x = -120/(2 × -2) = -120/(-4) = 30 Step 5: Therefore, selling 30 units maximizes the profit The answer is 30.