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

2. 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.

3. Example 3: Sports Application
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?
h(t) = –16t2 + v0t + h0
Write the general projectile function.
h(t) = –16t2 + 80t + 0
Substitute 80 for v0 and 0 for h0.

4. The ball will hit the ground when its height is zero.
Example 3 Continued
The ball will hit the ground when its height is zero.
–16t2 + 80t = 0
Set h(t) equal to 0.
–16t(t – 5) = 0
Factor: The GCF is –16t.
–16t = 0 or (t – 5) = 0
Apply the Zero Product Property.
t = 0 or t = 5
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.

5. 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) = –16t2 + v0t + h0
Write the general projectile function.
h(t) = –16t2 + 48t + 0
Substitute 48 for v0 and 0 for h0.

6. Check It Out! Example 3 Continued
The ball will hit the ground when its height is zero.
–16t2 + 48t = 0
Set h(t) equal to 0.
–16t(t – 3) = 0
Factor: The GCF is –16t.
–16t = 0 or (t – 3) = 0
Apply the Zero Product Property.
t = 0 or t = 3
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.

7. 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 x – 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?

8. Understand the Problem
Example 4 Continued 1
Understand the Problem
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 x – 4200.

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

10. Find the critical values by solving the related equation.
Example 4 Continued
Solve 3
Write the inequality.
–8x x – 4200 ≥ 6000
Find the critical values by solving the related equation.
–8x x – 4200 = 6000
Write as an equation.
–8x x – 10,200 = 0
Write in standard form.
–8(x2 – 75x ) = 0
Factor out –8 to simplify.

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

12. Example 4 Continued
Solve 3
Test an x-value in each of the three regions formed by the critical x-values.
Critical values
Test points

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

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

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

16. Check It Out! Example 4
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 x – The trip will be rescheduled if the profit is less $7500. How many people must have signed up if the trip is rescheduled?

17. Understand the Problem
Check It Out! Example 4 Continued 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 x – 5000.

18. Check It Out! Example 4 Continued2
Make a Plan
Write an inequality showing profit less than $7500. Then solve the inequality by using algebra.

19. Check It Out! Example 4 Continued
Solve 3
Write the inequality.
–25x x – 5000 < 7500
Find the critical values by solving the related equation.
–25x x – 5000 = 7500
Write as an equation.
–25x x – 12,500 = 0
Write in standard form.
–25(x2 – 50x + 500) = 0
Factor out –25 to simplify.

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

21. Check It Out! Example 4 Continued
Solve 3
Test an x-value in each of the three regions formed by the critical x-values.
Critical values
Test points

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

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

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