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?

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