+817 An aquarium is making a cylindrical shark tank. The volume must = 6000pi cubic ft. The top of the tank costs $1 per square foot, the glass siding 4$/cubic foot, and the cement bottom costs 2$ per square foot. Find the minimal cost.
+129850 Let r be the radius of the tank
The volume of the tank is
6000pi = pi * r^2 * h
Solving for h, we have that
h = 6000/r^2
So....the surface area cost is given by
Sa cost = ($1)pi*r^2 + ($2)pi *r^2 + ($4)2pi * r * 6000/r^2
Sa cost = 3pi*r^2 + 8pi* 6000r^(-1)
Take the derivative of this to find the radius that minimizes the cost
Sa ' = 6pir - 48000pi / r^2
Set this to 0 and solve for r
pi ( 6r - 48000/r^2) = 0
6r - 48000/^2 = 0
6 ( r - 8000/^2) = 0
r - 8000/r^2 = 0
r^3 - 8000 = 0
r^3 = 8000
r = 20 ft
So....the minimum cost is
3pi*(20)^2 + 8pi* 6000/20 =
pi [ 1200 + 2400] = $ 3600pi ≈ $11309.73
