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