Allie is playing basketball. She takes a shot 24 ft away. when the ball is 4 feet away from her, it is at a height of 10 feet above the floor. the ball reaches its greatest height of 18 ft above the floor when it is 12 feet away from her. a. find the value of a. b. if the hoop is 10 ft high, how close does Allie have to be in order to make the basket?

Part (a)

In decimal form, this is a = -0.125

-------------------

Explanation:

When it says "she takes the shot 24 feet away", that means that the horizontal distance from the basket to Allie is 24 feet. It's not measuring the slanted or diagonal distance directly from Allie's hand to the net itself. We can think of it as the distance from her feet to the base of the hoop.

When measuring the distance from Allie to the ball, we use the same idea. We simplify things using horizontal distance. Things get way too complicated if we measure the direct distance from the ball (wherever it may be in the air) to the net, or from the ball to Allie's hand.

x and y have units in feet.

The info "when the ball is 4 feet away from her, it is at a height of 10 feet above the floor" means we know the point (x,y) = (4,10) is on the parabola.

The other piece of info that "the ball reaches its greatest height of 18 ft above the floor when it is 12 feet away from her". This tells us the vertex is (h,k) = (12,18)

We have these four pieces of info

Plug those values into the equation below to isolate 'a'

y = a(x-h)^2 + k .... vertex form

10 = a(4-12)^2 + 18

10 = a(-8)^2 + 18

10 = 64a + 18

64a + 18 = 10

64a = 10-18

64a = -8

a = -8/64

a = -1/8

This converts to the decimal -1/8 = -0.125

============================================================

Part (b)

-------------------

Explanation:

From part (a), we found a = -1/8 = -0.125

This means that the vertex form equation y = a(x-12)^2 + 18 updates to y = -0.125(x-12)^2 + 18

This parabola represents the path the ball takes. Plug in y = 10 and solve for x. We do this because the hoop is 10 ft high, and we want to know what x values will make the equation true when y = 10. In other words, we want to know what two locations on the parabola in the form (x,10).

So,

y = -0.125(x-12)^2 + 18

10 = -0.125(x-12)^2 + 18

10-18 = -0.125(x-12)^2

-8 =  -0.125(x-12)^2

-0.125(x-12)^2 = -8

(x-12)^2 = -8/(-0.125)

(x-12)^2 = 64

x-12 = 8 or x-12 = -8

x = 8+12 or x = -8+12

x = 20 or x = 4

The two solutions are x = 4 and x = 20.

The two points we focus on are (4,10) and (20,10). This is when the ball is 10 ft off the ground. Earlier we were told that when the ball is 4 ft away from her, the ball is 10 ft off the ground. So that helps confirm why x = 4 works as a solution. We'll ignore this value though because we want the ball to come down through the hoop, not go up through the hoop. See the diagram below.

Therefore, she needs to be 20 ft away from the hoop to make the shot. Her being 24 ft away means that she's 24-20 = 4 ft too far.

Check out the graph below. We have the following points

As you can see in the diagram below, the green arc represents her shot if she's 24 ft away. She misses point D (the hoop). To make the hoop, she needs to move 4 ft closer to the basket. The red arc is the path the ball takes that will land in the hoop. Every point on the green arc has been shifted over 4 units to the right to arrive at a corresponding point on the red arc.

were is the equation