3.1 – Applications of Quadratics
Directions: Show all work to receive full credit. You may not use a graphing calculator to find the answer.
You may use a graphing calculator to check your final solution.
1. Determine whether the quadratic function 𝑓(𝑥) = 𝑥 2 − 4𝑥 − 5 has a maximum or minimum value.
Then find the maximum or minimum value.
2. The marketing department at Texas Instruments has found that, when certain calculators are sold at a
price of p dollars per unit, the revenue R (in dollars) as a function of the price p is
𝑅(𝑝) = −150𝑝2 + 21,000𝑝.
a. What unit price should be established to maximize revenue?
b. If this price is charged, what is the maximum revenue?
c. Graph the function to illustrate your solution. (Please use graph paper)
3. A farmer has 2000 yards of fence to enclose a rectangular field.
a. What are the dimensions (length and width) of the rectangle that encloses the most area?
b. Using these dimensions, what is the maximum area?
3.1 – Applications of Quadratics
c. Graph the function to illustrate your solution. (Please use graph paper)
4. A projectile is fired from a cliff 500 feet above the water at an inclination of 45 with a muzzle velocity
of 400 feet per second. In physics, it is established that the height of the projectile above the water is
−32𝑥 2
given by ℎ(𝑥) = (400)2 + 𝑥 + 500 where x is the horizontal distance of the projectile from the base of
the cliff.
a. Find the maximum height of the projectile
b. How far from the base of the cliff with the projectile strike the water?
c. Graph the function to illustrate your solutions. ( Please use graph paper)
5. A suspension bridge has towers 80 feet above the surface of the road and 500 feet apart. Cables attached
to the top of the towers form a parabolic curve that touches the road surface in the middle of the bridge.
Find the height of the cable 125 feet from the center of the bridge. (Assume the road is level)
3.1 – Applications of Quadratics
6. The John Deere company has found that the revenue from sales of heavy-duty tractors is a funciton of
1
the unit price p (in dollars) that it cahrges. Let the revenue R be 𝑅(𝑝) = − 2 𝑝2 + 1900𝑝.
a. What unit price p (in dollars) should be charged to maximize the revenue?
b. If this price is charged, what is the maximum revenue?
c. Graph the function to illustrate your solutions. (Please use graph paper)
7. Beth has 3000 feet of fencing available to enclose a rectangular field.
a. For what length is the area the largest?
b. What is the maximum area?
c. Graph the function to illustrate the solution. (Please use graph paper)
3.1 – Applications of Quadratics
8. A projectile is fired at an inclination of 45 with a muzzle velocity of 100 feet per second. The height h
−32𝑥 2
of the projectile is given by ℎ(𝑥) = (100)2 + 𝑥 where x is the horizontal distance of the projectile from
the firing point.
a. How far from the firing point is the height of the projectile a maximum?
b. Find the maximum height of the projectile.
c. How far from the firing point will the projectile strike the ground?
d. Graph the function to illustrate your solutions. (Please use graph paper)
9. A suspension bridge has towers 75 meters above the surface of the road and 400 meters apart. Cables
attached to the top of the towers form a parabolic curve that touches the road surface in the middle of the
bridge. Find the height of the cable 100 meters from the center of the bridge. (Assume the road is level)