Question

Oct 16, 10:30:54 AM

A rocket is shot into the air. The function f (x) = -16x2 + 64x + 8 gives the
height of the rocket (in feet) as a function of the rockets horizontal distance from
where it was initially shot.
a. What was the initial height of the rocket when it was shot?
b. What is the maximum height the rocket reaches in the air?
a. The initial height of the rocket was
feet.
b. The maximum height the rocket reaches is
feet.

Answer 1

A) 8 feet.

B) 72 feet

Explanation:

We have the function
`f(x)=-16x^2+64x+8` which gives the height of the rocket (in feet) as a function of the rocket's horizontal distance.

Part A)

We want to find the initial height of the rocket when it was shot.

At the initial height, the rocket has not moved anywhere. So, the horizontal distance will be 0.

Therefore, to find the initial height, we will substitute 0 into our function. This yields:

`f(0)=-16(0)^2+64(0)+8`

Evaluate:

`f(0)=8`

Therefore, the initial height was 8 feet.

Part B)

Notice that our function is a quadratic.

Therefore, the maximum height will be given by the vertex of our quadratic.

To find the vertex, we use:

`(-(b)/(2a),f(-(b)/(2a)))`

Let's label our coefficients. We have
`-16x^2+64x+8`

Therefore, a=-16, b=64, and c=8.

Substitute them into the vertex formula to find the x-coordinate:

`x=-(64)/(2(-16))\\\Rightarrow x=64/32=2`

Now, to find the maximum height, substitute 2 back into our function f(x):

`f(2)=-16(2)^2+64(2)+8`

Evaluate:

`f(2)=-16(4)+64(2)+8\\\Rightarrow f(2)=-64+128+8\\\Rightarrow f(2)=72\text{ feet}`

Therefore, the rocket reaches a maximum height of 72 feet.