Answer:
radius x = 3 ft
height h = 23,8 ft
Step-by-step explanation:
From problem statement
V(t) = V(cylinder) + V(hemisphere)
let x be radius of base of cylinder (at the same time radius of the hemisphere)
and h the height of the cylinder, then:
V(c) = π*x²*h area of cylinder = area of base + lateral area
A(c) = π*x² + 2*π*x*h
V(h) = (2/3)*π*x³ area of hemisphere A(h) = (2/3)*π*x²
A(t) = π*x² + 2*π*x*h + (2/3)*π*x²
Now A as fuction of x
total volume 505 = π*x²*h + (2/3)*π*x³
h = [505 - (2/3)* π*x³ ] /2* π*x
Now we have the expression for A as function of x
A(x) = 3π*x² + 2π*x*h A(x) = 3π*x² + 505 - (2/3)π*x³
Taking derivatives both sides
A´(x) = 6πx - 2πx² A´(x) = 0 6x - 2x² = 0
x₁ = 0 we dismiss
6 - 2x = 0
x = 3 and h = [505 - (2/3)* π*x³]/2* π*x
h = (505 - 18.84) / 6.28*3
h = 23,8 ft
![\bf \stackrel{f(x)}{0}=-4cos\left(x-\frac{\pi }{2} \right)\implies 0=cos\left(x-\frac{\pi }{2} \right) \\\\\\ cos^{-1}(0)=cos^{-1}\left[ cos\left(x-\frac{\pi }{2} \right) \right]\implies cos^{-1}(0)=x-\cfrac{\pi }{2} \\\\\\ x-\cfrac{\pi }{2}= \begin{cases} \frac{\pi }{2}\\\\ \frac{3\pi }{2} \end{cases}](https://tex.z-dn.net/?f=%5Cbf%20%5Cstackrel%7Bf%28x%29%7D%7B0%7D%3D-4cos%5Cleft%28x-%5Cfrac%7B%5Cpi%20%7D%7B2%7D%20%20%5Cright%29%5Cimplies%200%3Dcos%5Cleft%28x-%5Cfrac%7B%5Cpi%20%7D%7B2%7D%20%20%5Cright%29%0A%5C%5C%5C%5C%5C%5C%0Acos%5E%7B-1%7D%280%29%3Dcos%5E%7B-1%7D%5Cleft%5B%20cos%5Cleft%28x-%5Cfrac%7B%5Cpi%20%7D%7B2%7D%20%20%5Cright%29%20%5Cright%5D%5Cimplies%20cos%5E%7B-1%7D%280%29%3Dx-%5Ccfrac%7B%5Cpi%20%7D%7B2%7D%0A%5C%5C%5C%5C%5C%5C%0Ax-%5Ccfrac%7B%5Cpi%20%7D%7B2%7D%3D%0A%5Cbegin%7Bcases%7D%0A%5Cfrac%7B%5Cpi%20%7D%7B2%7D%5C%5C%5C%5C%0A%5Cfrac%7B3%5Cpi%20%7D%7B2%7D%0A%5Cend%7Bcases%7D)
