Problem 35 - Essential Calculus

Question

A solid is formed by adjoining two hemispheres to the ends of a right circular cylinder. The total volume of the solid is 12 cubic centimeters. Find the radius of the cylinder that produces the minimum surface area.

Step-by-Step Solution

Answer

To find the exact value for the radius, we need to solve the equation \(-24 - \frac{8}{3}r^2 + 8\pi r^3 = 0\). The solution will provide the radius of the cylinder that produces the minimal surface area for the solid.

Step 1: Formulate the volume and surface area equations

The volume \(V\) of this shape is the sum of the volume of the cylinder \(V_c = \pi r^2h\) and the volumes of the two hemispheres \(V_h = 2*\frac{2}{3}\pi r^3\). Therefore, \(V= \pi r^2h + \frac{4}{3}\pi r^3\). Since it's given that \(V=12\), we have \(12= \pi r^2h + \frac{4}{3}\pi r^3\). We can express \(h\) in terms of \(r\): \(h = \frac{12 - \frac{4}{3}\pi r^3}{\pi r^2}\). Now, the surface area \(A\) of the shape is the sum of the surface area of the cylinder \(A_c = 2\pi rh\) and the surface areas of the two hemispheres \(A_h = 2*2\pi r^2\). Therefore, \(A = 2\pi rh + 4\pi r^2\).

Step 2: Substitute \(h\) into the surface area equation

Substitute the \(h\) obtained in step 1 into the surface area equation. We obtain \(A = 2\pi r * \frac{12 - \frac{4}{3}\pi r^3}{\pi r^2} + 4\pi r^2 = \frac{24}{r} - \frac{8}{3}r + 4\pi r^2\).

Step 3: Find the minimum surface area

To find the minimum surface area, we need to find the derivative of \(A\) with respect to \(r\), set the derivative equal to zero and then solve for \(r\). \(A'\) is given by \(A' = -\frac{24}{r^2} - \frac{8}{3} + 8\pi r\). Setting \(A' = 0\), we get \(-\frac{24}{r^2} - \frac{8}{3} + 8\pi r = 0\). To solve this equation, multiply it throughout by \(r^2\) to get \(-24 - \frac{8}{3}r^2 + 8\pi r^3 = 0\). The roots of this equation will give us the possible values of \(r\). We then take the second derivative \(A''\) and substitute the root values to find which gives a minimum.

Step 4: Find the root(s) for the minimum surface area

The roots of the equation can be found using any numerical root finding method such as the bisection method, Newton's method, or trial and error. After finding the roots, find the second derivative \(A''\) and substitute the roots. The root value giving a positive \(A''\) will be our minimum surface area.