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
