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 inches. Find the radius of the cylinder that produces the minimum surface area.

Short answer

Upon completing steps 1 to 4, we will find the radius of the cylinder that produces the least surface area for the solid whose volume is given as 12 cubic inches.

Step-by-step solution

Write down the formulas of volume and surface area

The volume of a cylinder is given by \(\pi r^2h\) and the volume of a sphere by \(\frac{4}{3}\pi r^3\). Therefore, the total volume of the solid (cylinder + 2 hemispheres) is: \[V = \pi r^2h + 2 (\frac{1}{2} \cdot \frac{4}{3}\pi r^3) = \pi r^2h + \frac{4}{3}\pi r^3\] Substitute for \(h\) from the volume equation, since we know the total volume is 12 cubic inches.

Find the equation for the surface area of the solid

The total surface area (A) of the solid is given by the sum of the surface areas of the cylinder (2πrh) and the two hemispheres (2πr^2 each, adding up to 4πr^2). Thus, \[A = 2\pi rh + 4\pi r^2\]. Substitute for \(h\) from the volume equation obtained in Step 1, we get the surface area in terms of radius only.

Find derivative of A in respect to r

Take the derivative of the total surface area function with respect to \(r\). Set it equal to 0 and solve for \(r\) to find the value that minimizes the surface area.

Verify the minimum value

Finally, find the second derivative of the surface area function at \(r\) and check if it’s positive to ensure it’s a minimum point.