Answer:
Option D - 64 feet.
Step-by-step explanation:
Given : Ja'Von kicks a soccer ball into the air. The function [tex]f(x) = -16(x-2)^2 + 64[/tex] represents the height of the ball, in feet, as a function of time, x, in seconds.
To find: What is the maximum height the ball reaches?
Solution :
To find the maximum height of the ball we find the derivative,
[tex]f(x) = -16(x-2)^2 + 64[/tex]
Derivate w.r.t x
[tex]f'(x) = -16\times 2(x-2)+0[/tex]
[tex]f'(x) = -32(x-2)[/tex]
To find maxima put [tex]f'(x)=0[/tex]
[tex]0= -32(x-2)[/tex]
[tex]32x=64[/tex]
[tex]x=2[/tex]
Now, we find the second derivative form maximum,
[tex]f''(x) = -32<0[/tex]
Therefore, The maximum point is at x=2
Substitute x=2 in the given function,
[tex]f(2) = -16(2-2)^2 + 64[/tex]
[tex]f(2) = -16(0)+ 64[/tex]
[tex]f(2) = 64[/tex]
Therefore, Option D is correct.
The maximum height the ball reaches is 64 feet.
