word question

Question

 Given a closed circular cylinder with volume 120 cubic feet and given cost per surface area foot of $.75 for the top and bottom and $.5 for the sides, determine the dimensions, both radius and height, and cost that minimize the surface area cost

Answer 1

The volume of a cylinder is     V = πr²h

 

The surface area of a cylinder is     A = 2πr² + 2πrh

 

 

Using these formulas, we can write equations to represent the volume and cost.

 

120 = πr²h                     eq1

 

C = 0.75πr² + 0.75πr² + 0.5(2πrh)

 

C = 1.5πr² + πrh                 eq2

 

 

Substitute eq1 into eq2 to get eq2 in terms of r.

 

C = 1.5πr² + πr(120 / πr²)

 

C = 1.5πr² + 120r^-1

 

 

Next, we set the derivative of C equal to zero.

 

d/dx(C) = 0

 

3πr - 120r^-2 = 0

 

3πr - (120 / r²) = 0

 

(3πr³ - 120) / r² = 0

 

 

Set the numerator equal to zero and solve for r.

 

3πr³ - 120 = 0

 

3πr³ = 120

 

 r³ = 12.7324

 

 r = 2.3351

 

 

Now we perform test points.  We evaluate the derivative of C at r=2 and r=3.

 

C(2) = (3π(2³) - 120) / (2²)

 

        = (24π - 120) / 4

        

        = -44.60 / 4

       

        = -11.15

 

C(3) = (3π(3³) - 120) / (3²)

 

        = (81π - 120) / 9

 

        = 134.46 / 9

 

         = 14.941 

 

 

Since the derivative changes from negative to positive, we have a minimum at r=2.3351

 

 

Evaluate the height at this radius.

 

h = 120 / π(2.3351)²

 

h = 7

 

 

So a radius of 2.3351 feet and a height of 7 feet will minimize the surface area cost.