A soup company wants to manufacture a can in the shape of a right circular cylinder that will hold 500 cm^3 of liquid. The material for the top and bottom costs 0.02 cent/cm^2, and the material for the sides costs 0.01 cent/cm^2.

a) Estimate the radius r and height h of the can that costs the least to manufacture. [suggestion: Express the cost C in terms of r.]

The area of the top and bottom:

2πr²

Cost for top and bottom:

2πr² x 0.02

= 0.04πr²

Area for side:

2πrh

Cost for side:

2πrh x 0.01

= 0.02πrh

Total cost:

C = 0.04πr² + 0.02πrh

We know that the volume of the can is:

V = πr²h

h = 500/πr²

Substituting this into the cost equation to get a cost function of radius:

C (r) = 0.04πr² + 0.02πr (500/πr²)

C (r) = 0.04πr² + 10/r

Now, we differentiate with respect to r and equate to 0 to obtain the minimum value:

0 = 0.08πr - 10/r²

10/r² = 0.08πr

r³ = 125/π

r = 3.41 cm