Chapter 3.4, Problem 11AE

Analyzing the Motion of a Projectile A projectile is fired from a cliff 200 feet above the water at an inclination of ${45}^{\circ }$ to the horizontal, with a muzzle velocity of 50 feet per second. The height $h$ of the projectile above the water is modeled by

$h\left(x\right)=\frac{-32x^2}{{\left(50\right)}^2}+x+200$

where $x$ is the horizontal distance of the projectile from the face of the cliff.

(a) At what horizontal distance from the face of the cliff is the height of the projectile a maximum?

(b) Find the maximum height of the projectile.

(c) At what horizontal distance from the face of the cliff will the projectile strike the water?

(d) Using a graphing utility, graph the function $h$, $0\le x\le 200$.

(e) Use a graphing utility to verify the solutions found in parts (b) and (c).

(f) When the height of the projectile is 100 feet above the water, how far is it from the cliff?