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.

Landry Dec 10, 2018

#1**+2 **

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

CPhill Dec 10, 2018