Need help asap. Quadratic Func.

Question

The height, h (in feet), of a golf ball depends on the time, t (in seconds), it has been in the air. Sarah hits a shot off the tee that has a height modeled by the quadratic function f(h) = -16(t – 2.5)² + 100.

1. What is the maximum height of the golf ball? How long does it take for the golf ball to reach the maximum height?

2. What is the total flight time of the ball?

3. What is the domain of the function as it is applied to the actual situation? Make sure you use correct notation.

4. Suppose that the tee location was moved to a spot 20 feet higher up a hill. Sarah hit the exact same shot from that location. What effect does this change in height have on the quadratic function compared to the original model?

Answer 1

*** one note here about the question - if f(h) is meant to measure height, it should be f(t). Height is a function of the amount of time the ball is in the air, not the other way around. Height is the dependent or (y) variable here, and time is the independent or (x) variable here). We can still solve this, but it's worth asking your teacher about why the equation is written this way.

1.

1. f(h) will be at its maximum when the the curve of this graph reaches its vertex. So, we'll identify values for a, b, and c, and solve using the quadratic formula.

f(h) = -16(t^2 - 5t + 6.25) + 100 ----- reorganize your terms into ax^2 + bx + c notation.

f(h) = -16t^2 +80t - 100 + 100 = 0

a = -16, b = 80, c = 0

1. Now let's use the vertex formula: x = -b/2a
2. -80/(2*-16) = 2.5
3. 2.5 is the x coordinate of the vertex, which in this case is the time that the ball has been in the air when it reaches it's maximum height. We can now solve for the y coordinate, or the height of the ball at this vertex, by plugging in 2.5s to our original equation.

Max. height = -16(2.5 - 2.5)^2 + 100 ------- (solve algebraically)

Max. height = 100

2.

1. Thinking about this equation graphically, height will be equal to 0 when time = 0 (start) and f(h) will be equal to 0 when the ball comes back down to the ground. We don't know how long it will take for the ball to get back down to the ground, but we know that f(h) will be equal to zero. So let's solve by setting f(h) equal to 0.

0 = -16(t-2.5)^2 + 100

16(t - 2.5)^2 = 100 (take the square root of both sides)

4(t - 2.5) = 10 (distribute the 4)

4t - 10 = 10 (solve algebraically)

4t = 20

t = 5

1. Now we know that total time that the ball is in the air: 5 seconds.

3.

1. The domain here is the amount of time the ball is in the air. The function is constrained by time 0s to 5s.

4.

1. Assuming that everything else in this equation is the same, and that the tee is 20 feet higher but not the fairway, this means that the ball will fall slightly longer than it rises. If she hits the same shot, the max height will simply be exactly 20 feet higher (due to the 20 foot increase in starting height). The other thing that will change, is that the ball will be in the air slightly longer.