Quadratic Application
ObjectivesI can solve real life situations
represented by quadratic equations.
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.
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) = –16t2 + v0t + h0
h(t) = –16t2 + 80t + 0
Write the general projectile function.
Substitute 80 for v0 and 0 for h0.
Example 3 Continued
The ball will hit the ground when its height is zero.
–16t2 + 80t = 0–16t(t – 5) = 0–16t = 0 or (t – 5) = 0t = 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.
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
h(t) = –16t2 + 48t + 0
Write the general projectile function.
Substitute 48 for v0 and 0 for h0.
Check It Out! Example 3 Continued
The ball will hit the ground when its height is zero.
–16t2 + 48t = 0–16t(t – 3) = 0–16t = 0 or (t – 3) = 0t = 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.
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) = –8x2 + 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?
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) = –8x2 + 600x – 4200.
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
Solve3
Write the inequality.–8x2 + 600x – 4200 ≥ 6000
–8x2 + 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.
–8x2 + 600x – 10,200 = 0
–8(x2 – 75x + 1275) = 0
Example 4 Continued
Solve3
Use the Quadratic Formula.
Simplify.
x ≈ 26.04 or x ≈ 48.96
Example 4 Continued
Solve3
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
Solve3
–8(25)2 + 600(25) – 4200 ≥ 6000
–8(45)2 + 600(45) – 4200 ≥ 6000
–8(50)2 + 600(50) – 4200 ≥ 6000
5800 ≥ 6000Try 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
Solve3
For a profit of $6000, the average price of a helmet needs to be between $26.04 and $48.96, inclusive.
Example 4 Continued
Look Back4
Enter y = –8x2 + 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
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) = –25x2 + 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
1 Understand the ProblemThe 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) = –25x2 + 1250x – 5000.
Check It Out! Example 4 Continued
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
Solve3
Write the inequality.
–25x2 + 1250x – 5000 < 7500
–25x2 + 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.
–25x2 + 1250x – 12,500 = 0
–25(x2 – 50x + 500) = 0
Check It Out! Example 4 Continued
Simplify.
x ≈ 13.82 or x ≈ 36.18
Use the Quadratic Formula.
Solve3
Check It Out! Example 4 Continued
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
Solve3
Check It Out! Example 4 Continued
–25(13)2 + 1250(13) – 5000 < 7500
7025 < 7500Try 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
Solve3
Check It Out! Example 4 Continued
The trip will be rescheduled if the number of people signed up is fewer than 14 people or more than 36 people.
Solve3
Check It Out! Example 4 Continued
Look Back4
Enter y = –25x2 + 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