Problem 33 - Calculus of a Single Variable

Question

Minimum Surface Area A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 14 cubic centimeters. Find the radius of the cylinder that produces the minimum surface area.

Step-by-Step Solution

Answer

The radius of the cylinder that produces the minimum surface area is 3.36 cm.

Step 1: Identify and Analyze the Variables

Let \( r \) be the radius of the hemispheres and the cylinder, and \( h \) be the height of the cylinder alone. The total volume is the sum of the volumes of the two hemispheres and the cylinder, which should equal to 14cm3. So we have \(2*(2/3)*\pi*r^3 + \pi*r^2*h = 14\).

Step 2: Express h in terms of r

From the equation in step 1, solve for \( h \) in terms of \( r \): \( h = (14 - 4/3*\pi*r^3)/(\pi * r^2) \).

Step 3: Formulate the Surface Area

The surface area S of the solid is the sum of the surface areas of the two hemispheres and the lateral area of the cylinder: \( S = 2*(2*\pi*r^2) + 2*\pi*r*h \). Substituting for \( h \) gives \( S = 4 * \pi * r^2 + 2 * \pi * r * ((14 - 4/3 * \pi * r^3)/(\pi * r^2)) \). Simplify this to \( S = 4 * \pi * r^2 + (28 * r - 8/3 * r) \).

Step 4: Solve for Minimum Surface Area

To find the minimum surface area, take the derivative of \( S \) with respect to \( r \), set it equal to 0 and solve for \( r \). This gives \( S' = 8 * \pi * r - 28 + 8/3 \). Solving for \( r \) gives \( r = 3.36 cm \). Verify this is a minimum by checking the second derivative.

Step 5: Verify the Solution

The second derivative of \( S \) with respect to \( r \) is \( S'' = 8 * \pi \). Since this value is positive, it confirms that \( r = 3.36 cm \) is indeed the radius of the cylinder that produces the minimum surface area.