You have been asked to design a can with a volume of 672cm3 that is shaped like a right circular cylinder. The can will have a closed top. What radius r and height h, in centimeters, would minimize the amount of material needed to construct this can

r = 4.747 cm and h = 9.4925 cm

Explanation:

We know that volume of a cylinder is given as:

V = πr²h

Also, surface area is given as;

S = 2πr² + 2πrh

Where r is radius and h is height

Now, we are told that the volume is 672 cm³

Thus, πr²h = 672

Making h the subject gives;

h = 672/πr²

Putting 672/πr² for h in the surface area equation gives;

S = 2πr² + 2πr(672/πr²)

Factorizing gives;

S = 2π[r² + 672/πr]

Differentiating to get first derivative gives;

S' = 2π[2r - (672/πr²)]

Equating to zero gives;

2π[2r - (672/πr²)] = 0

4πr - 1344/r² = 0

4πr = 1344/r²

r³ = 1344/4π

r³ = 106.95212175775

r = ∛106.95212175775

r = 4.747 cm

So, since h = 672/πr²

Then, h = 672/π(4.747)²

h = 9.4925 cm