Answer to Question #258469 in Calculus for Anne

Question

Express the height of a cylindrical can as a function of the diameter, if 18pi sq.in. of sheet metal is used. H=?

Answer 1

The total surface area of a solid is the sum of the areas of all of its faces. A cylinder has 3 faces:

the top and bottom circular bases, and the lateral surface of the cylinderю

If the diameter of a circular base is $D,$ then the area of each base must be:

$$A_{base}=\dfrac{\pi D^2}{4}$$

$$A_{lateral}=2\pi(\dfrac{D}{2})H=\pi DH$$

$$A_{total}=2A_{base}+A_{lateral}=\dfrac{\pi D^2}{4}+\pi DH$$

If $18\pi$ sq.in. of sheet metal is used

$$\dfrac{\pi D^2}{4}+\pi DH=18\pi$$

$$H=H(D)=\dfrac{18}{D}-\dfrac{D}{4}$$

$$H(D)=\dfrac{18}{D}-\dfrac{D}{4} , inches$$