A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. An industrial tank of this shape must have a volume of 4640 cubic feet. The hemispherical ends cost twice as much per square foot of surface area as the sides. Find the dimensions that will minimize the cost. (Round your answers to three decimal places.)

The dimensions that minimize the cost of building the tank are .

Let

and

and

Therefore,

From the question

The total cost becomes

We need to eliminate . The volume from the question gives a way out

substitute into the formula for total cost gives, after simplifying

differentiating with respect to , we get

at extrema

To confirm that is a minimum value, carry out the second derivative test

substituting , we get that > , confirming that minimum value

To find , recall that

substituting , we get as the corresponding minimum height

Therefore, minimize the total cost of building the tank.

Learn more about minimizing dimensions to reduce costs here: link

Radius =6.518 feet

Height = 26.074 feet

Step-by-step explanation:

The Volume of the Solid formed  = Volume of the two Hemisphere + Volume of the Cylinder

Volume of a Hemisphere  

Volume of a Cylinder

Therefore:

The Volume of the Solid formed

Area of the Hemisphere =

Curved Surface Area of the Cylinder =

Total Surface Area=

Cost of the Hemispherical Ends  = 2 X  Cost of the surface area of the sides.

Therefore total Cost, C

Recall:

Therefore:

The minimum cost occurs at the point where the derivative equals zero.

Recall:

Therefore, the dimensions that will minimize the cost are:

Radius =6.518 feet

Height = 26.074 feet