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Quadratic Application Objectives I can solve real life situations represented by quadratic equations.

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Any object that is thrown or launched into the air, such as a baseball, basketball, or soccer ball, is a projectile. The general function that approximates the height h in feet of a projectile on Earth after t seconds is given. Note that this model has limitations because it does not account for air resistance, wind, and other real-world factors.

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A golf ball is hit from ground level with an initial vertical velocity of 80 ft/s. After how many seconds will the ball hit the ground? Example 3: Sports Application h(t) = –16t 2 + v 0 t + h 0 h(t) = –16t 2 + 80t + 0 Write the general projectile function. Substitute 80 for v 0 and 0 for h 0.

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Example 3 Continued The ball will hit the ground when its height is zero. –16t 2 + 80t = 0 –16t(t – 5) = 0 –16t = 0 or (t – 5) = 0 t = 0 or t = 5 Set h(t) equal to 0. Factor: The GCF is –16t. Apply the Zero Product Property. Solve each equation. The golf ball will hit the ground after 5 seconds. Notice that the height is also zero when t = 0, the instant that the golf ball is hit.

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Check It Out! Example 3 A football is kicked from ground level with an initial vertical velocity of 48 ft/s. How long is the ball in the air? h(t) = –16t 2 + v 0 t + h 0 h(t) = –16t 2 + 48t + 0 Write the general projectile function. Substitute 48 for v 0 and 0 for h 0.

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Check It Out! Example 3 Continued The ball will hit the ground when its height is zero. –16t 2 + 48t = 0 –16t(t – 3) = 0 –16t = 0 or (t – 3) = 0 t = 0 or t = 3 Set h(t) equal to 0. Factor: The GCF is –16t. Apply the Zero Product Property. Solve each equation. The football will hit the ground after 3 seconds. Notice that the height is also zero when t = 0, the instant that the football is hit.

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Example 4: Problem-Solving Application The monthly profit P of a small business that sells bicycle helmets can be modeled by the function P(x) = –8x 2 + 600x – 4200, where x is the average selling price of a helmet. What range of selling prices will generate a monthly profit of at least $6000?

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1 Understand the Problem Example 4 Continued The answer will be the average price of a helmet required for a profit that is greater than or equal to $6000. List the important information: The profit must be at least $6000. The function for the business’s profit is P(x) = – 8x 2 + 600x – 4200.

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2 Make a Plan Write an inequality showing profit greater than or equal to $6000. Then solve the inequality by using algebra. Example 4 Continued

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Solve 3 Write the inequality. –8x 2 + 600x – 4200 ≥ 6000 –8x 2 + 600x – 4200 = 6000 Find the critical values by solving the related equation. Write as an equation. Write in standard form. Factor out –8 to simplify. –8x 2 + 600x – 10,200 = 0 –8(x 2 – 75x + 1275) = 0 Example 4 Continued

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Solve 3 Use the Quadratic Formula. Simplify. x ≈ 26.04 or x ≈ 48.96 Example 4 Continued

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Solve 3 Test an x-value in each of the three regions formed by the critical x-values. 10 20 30 40 50 60 70 Critical values Test points Example 4 Continued

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Solve 3 –8(25) 2 + 600(25) – 4200 ≥ 6000 –8(45) 2 + 600(45) – 4200 ≥ 6000 –8(50) 2 + 600(50) – 4200 ≥ 6000 5800 ≥ 6000 Try x = 25. Try x = 45. Try x = 50. 6600 ≥ 6000 5800 ≥ 6000 Write the solution as an inequality. The solution is approximately 26.04 ≤ x ≤ 48.96. x x Example 4 Continued

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Solve 3 For a profit of $6000, the average price of a helmet needs to be between $26.04 and $48.96, inclusive. Example 4 Continued

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Look Back 4 Enter y = –8x 2 + 600x – 4200 into a graphing calculator, and create a table of values. The table shows that integer values of x between 26.04 and 48.96 inclusive result in y-values greater than or equal to 6000. Example 4 Continued

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A business offers educational tours to Patagonia, a region of South America that includes parts of Chile and Argentina. The profit P for x number of persons is P(x) = –25x 2 + 1250x – 5000. The trip will be rescheduled if the profit is less $7500. How many people must have signed up if the trip is rescheduled? Check It Out! Example 4

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1 Understand the Problem The answer will be the number of people signed up for the trip if the profit is less than $7500. List the important information: The profit will be less than $7500. The function for the profit is P(x) = –25x 2 + 1250x – 5000. Check It Out! Example 4 Continued

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2 Make a Plan Write an inequality showing profit less than $7500. Then solve the inequality by using algebra. Check It Out! Example 4 Continued

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Solve 3 Write the inequality. –25x 2 + 1250x – 5000 < 7500 –25x 2 + 1250x – 5000 = 7500 Find the critical values by solving the related equation. Write as an equation. Write in standard form. Factor out –25 to simplify. –25x 2 + 1250x – 12,500 = 0 –25(x 2 – 50x + 500) = 0 Check It Out! Example 4 Continued

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Simplify. x ≈ 13.82 or x ≈ 36.18 Use the Quadratic Formula. Solve 3 Check It Out! Example 4 Continued

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Test an x-value in each of the three regions formed by the critical x-values. 5 10 15 20 25 30 35 Critical values Test points Solve 3 Check It Out! Example 4 Continued

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–25(13) 2 + 1250(13) – 5000 < 7500 7025 < 7500 Try x = 13. Try x = 30. Try x = 37. 10,000 < 7500 7025 < 7500 Write the solution as an inequality. The solution is approximately x > 36.18 or x < 13.82. Because you cannot have a fraction of a person, round each critical value to the appropriate whole number. x –25(30) 2 + 1250(30) – 5000 < 7500 –25(37) 2 + 1250(37) – 5000 < 7500 Solve 3 Check It Out! Example 4 Continued

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The trip will be rescheduled if the number of people signed up is fewer than 14 people or more than 36 people. Solve 3 Check It Out! Example 4 Continued

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Look Back 4 Enter y = –25x 2 + 1250x – 5000 into a graphing calculator, and create a table of values. The table shows that integer values of x less than 13.81 and greater than 36.18 result in y-values less than 7500. Check It Out! Example 4 Continued

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