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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.

 Dec 10, 2018
 #1
avatar+98102 
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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

 

 

cool cool cool

 Dec 10, 2018

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