The problem asks us to find the radius and height of a right circular cylinder that has a capacity (volume) of 32 in³ and is constructed using the least amount of metal, meaning it aims to minimize the surface area.

Key formulas involved:

1. Volume of a cylinder:
   $V = \pi r^2 h$ where $r$ is the radius, $h$ is the height, and $V = 32 \, \text{in}^3$.

2. Surface area of a cylinder (the amount of metal used):
   $A = 2\pi r^2 + 2\pi r h$ where the first term represents the area of the two circular ends, and the second term is the area of the curved side.

Steps to solve:

1. Use the volume equation to express height $h$ in terms of radius $r$:
   From the volume equation: $h = \frac{V}{\pi r^2} = \frac{32}{\pi r^2}$

2. Substitute this into the surface area formula:
   $A = 2\pi r^2 + 2\pi r \left( \frac{32}{\pi r^2} \right) = 2\pi r^2 + \frac{64}{r}$

3. Minimize the surface area by taking the derivative of $A$ with respect to $r$ and setting it equal to 0: $\frac{dA}{dr} = 4\pi r - \frac{64}{r^2}$ Set this equal to zero to find the critical points: $4\pi r = \frac{64}{r^2}$ Multiply both sides by $r^2$: $4\pi r^3 = 64$ Solve for $r$: $r^3 = \frac{64}{4\pi} = \frac{16}{\pi}$ $r = \sqrt[3]{\frac{16}{\pi}}$

4. Find the corresponding height $h$: Once we have $r$, substitute it back into the equation for height: $h = \frac{32}{\pi r^2}$

Let me compute the exact values for you.The radius and height of the container that minimizes the amount of metal are approximately:

• Radius: $1.72 \, \text{in}$
• Height: $3.44 \, \text{in}$

Would you like further details or have any questions?

Here are some related questions to explore:

1. What is the total surface area of the cylinder with these dimensions?
2. How does changing the volume affect the optimal dimensions of the cylinder?
3. Could the container's shape become more cost-effective with a different shape than a cylinder?
4. How is the optimization process affected if we aim to minimize volume instead of surface area?
5. How would the dimensions change if the container is made for a different material with specific constraints?

Tip: When optimizing for cost, always consider how changing one dimension affects the others through constraints like volume.