Question

You are to construct a€cylindrical can out of tin that encloses a volume of exactly 100 cubic inches. to three significant figures, which dimensions (r and h) will give you the tin can that requires the least amount of metal?

Answer 1

The formula for volume of cylinder is:

V = π r^2 h

since volume is 100 cubic inches, so:

100 = π r^2 h

Rewriting in terms of h:

h = 100 / π r^2

The least amount of metal means least amount of surface area needed. The surface area of cylinder is:

A = 2 π r h + 2 π r^2

insert the value of h in terms of r:

A = 2 π r (100 / π r^2) + 2 π r^2

A = 200 r-1 + 2 π r^2

Take the 1st derivate and set to 0, dA/dr = 0:

dA/dr = - 200 r-2 + 4 π r

0 = - 200 r-2 + 4 π r

200 / r^2 = 4 π r

r^3 = 50 / π

r = 2.52 inches

So height h is:

h = 100 / π r^2

h = 100 / π (2.52)^2

h = 5.01 inches

Answers:

r = 2.52 inches

h = 5.01 inches