Question

A soup can in the shape of a right circular cylinder is to be made from two materials. The material for the side of the can costs $0.015 per square inch and the material for the lids costs $ 0.027 per square inch. Suppose that we desire to construct a can that has a volume of 16 cubic inches. What dimensions minimize the cost of the can?

a. Draw a picture of the can and label its dimensions with appropriate variables.
b. Use your variables to determine expressions for the volume, surface area, and cost of the can.
c. Determine the total cost function as a function of a single variable. What is the domain on which you should consider this function?
d. Find the absolute minimum cost and the dimensions that produce this value.

Answer 1

a) file annex

b) V(c) = π*x²*y

A(x) = 2*π*x² + 32/x

C(x) = 0,1695*x² + 0,48 /x

Domain C(x) = {x/ x >0}

d) C(min) = 0,64 $

x = 1.123 in radius of base

y = 4,04 in height of the can

Explanation:

See annex file

Lets:

call x = radius of the base of the cylinder and

y = the height of the cylinder

Then

Volume of the cylinder ⇒ V(c) = π*r²*h ⇒V(c) = π*x²*y

And y = V / ( π*x²) ⇒ V = 16 / ( π*x²)

Area of cylinder = lids area + lateral area

lids area = 2*π*x² ⇒ lateral area = 2*π*x*y

lateral area =2*π*x [16/(π*x²) ] ⇒ lateral area = 32/x

Then

A(x) = 2*π*x² + 32/x

Function cost C(x)

C(x) = 0.027 * 2*π*x² + 0.015 * (32/x)

C(x) = 0,1695*x² + 0,48 /x

Domain C(x) = {x/ x >0}

Now function cost:

C(x) = 0,1695*x² + 0,48 /x

Taking derivative:

C´(x) = 2*0,1695*x - 0.48/x² C´(x) = 0,339*x - 0.48/x²

C´(x) = 0 0.339*x³ - 0.48 = 0 x³ = 0.48/0.339 x³ = 1.42

x = 1.123 in

y = 16/πx² ⇒ y = 4,04 in

C(min) = 0,64 $