Question

The height of a batted ball is modeled by the function h=-0.01x^2+1.22x+3, where x is the horizontal distance in feet from the point of impact with the bat, and h is the height of the ball in feet. a) What is the maximum height that the ball will reach show work. b) At what distance from the batter will the ball be at its maximum height?

Answer 1

Part A:

Given that the height of a batted ball is modeled by the function $h=-0.01x^2+1.22x+3$, where x is the horizontal distance in feet from the point of impact with the bat, and h is the height of the ball in feet.

At maximum height,

$\frac{dh}{dx} =0$

$\frac{dh}{dt} =0 \\ \\ \Rightarrow-0.02x+1.22=0 \\ \\ \Rightarrow0.02x=1.22 \\ \\ \Rightarrow x= \frac{1.22}{0.02} =61$

Thus, the maximum height occurs when x = 61 feet and the maximum height is given by

$h=-0.01(61)^2+1.22(61)+3 \\ \\ =-0.01(3,721)+74.42+3=-37.21+77.42 \\ \\ =40.21$

Therefore, the maximum height is 40.21 feet.



Part B:

From part A, we can see that the maximum height occured when x = 61 feet.

Therefore, the distance from the batter the ball will be at its maximum height is 61 feet.