Question

You are asked to build an open cylindrical can (i.e. no top) that will hold 364.5 cubic inches. To do this, you will cut its bottom from a square of metal and form its curved side by bending a rectangular sheet of metal.(a) Express the total amount of material required for the square and the rectangle in terms of r. A(r) = __________(b) Find the radius and height of the can that will minimize the total amount of material required.r = _____h = _____

Answer 1

(a) A(r) = 4r² +729/r

(b) r = 4.5 inches; h = 5.73 inches

Explanation:

(a) For a given radius, the volume is given by the formula ...

V = πr²h

so the height is ...

h = V/(πr²)

Then the area of the rectangle required to form the curved side is ...

A = 2πrh = (2πr)(V/(πr²)) = 2V/r

The area required for the bottom is a square that is 2r on each side, so is ...

A = (2r)² = 4r²

The total area of required material is ...

A(r) = 4r² +2V/r

A(r) = 4r² +729/r

__

(b) The material will be minimized when the derivative of A(r) with respect to r is zero:

A' = 0 = 8r -2V/r²

2V/r² = 8r

V/4 = r³

r = ∛(V/4) = ∛(364.5/4) ≈ 4.5 . . . inches

h = (364.5)/(π·4.5²) = 18/π ≈ 5.73 . . . inches

Material will be minimized for a radius of 4.5 inches, and a height of 5.73 inches.