To determine the dimensions of an aluminum can that holds 320 in³ of juice and uses the least material, we need to minimize the surface area of the can while ensuring that it holds the given volume.

Assumptions:

• The can is cylindrical, which is a common shape for beverage cans.

• The formula for the volume $V$ of a cylinder is: $V = \pi r^2 h$ where $r$ is the radius of the base and $h$ is the height.

• The formula for the surface area $A$ of a cylinder (including the top, bottom, and side) is: $A = 2\pi r^2 + 2\pi rh$ where the first term represents the areas of the top and bottom circles, and the second term is the lateral surface area.

Given:

• Volume $V = 320 \, \text{in}^3$

We need to minimize $A$, the surface area, for the fixed volume. So, we have the following steps:

Step 1: Express height in terms of radius

Using the volume equation, solve for $h$: $320 = \pi r^2 h$ $h = \frac{320}{\pi r^2}$

Step 2: Substitute $h$ into the surface area formula

Now, substitute $h = \frac{320}{\pi r^2}$ into the surface area equation: $A = 2\pi r^2 + 2\pi r \left( \frac{320}{\pi r^2} \right)$ Simplify the second term: $A = 2\pi r^2 + \frac{640}{r}$

Step 3: Minimize the surface area

To minimize the surface area, take the derivative of $A$ with respect to $r$ and set it equal to 0: $\frac{dA}{dr} = 4\pi r - \frac{640}{r^2}$ Set this equal to 0: $4\pi r = \frac{640}{r^2}$ Multiply both sides by $r^2$: $4\pi r^3 = 640$ Solve for $r$: $r^3 = \frac{640}{4\pi}$ $r^3 = \frac{160}{\pi}$ $r = \left( \frac{160}{\pi} \right)^{\frac{1}{3}}$ Using a calculator to find the value: $r \approx 4.35 \, \text{in}$

Step 4: Calculate the height

Now that we have the radius $r$, substitute it back into the equation for height: $h = \frac{320}{\pi r^2}$ $h = \frac{320}{\pi (4.35)^2}$ $h \approx 5.37 \, \text{in}$

Final Dimensions:

• Radius $r \approx 4.35 \, \text{in}$
• Height $h \approx 5.37 \, \text{in}$

Conclusion:

The optimal dimensions for the aluminum can that holds 320 in³ of juice and uses the least material are approximately:

• Radius: 4.35 inches
• Height: 5.37 inches

Would you like more details or have any questions about this solution?

Related Questions:

1. What is the process for deriving the surface area formula for a cylinder?
2. How do you find the volume of different 3D shapes like spheres or cones?
3. Why is minimizing surface area important in packaging design?
4. How does the material used in packaging affect the total cost of manufacturing?
5. What is the relationship between volume and surface area in optimization problems?

Tip:

When solving optimization problems, always begin by expressing the variables you're trying to optimize in terms of each other using the given constraints. This simplifies the problem and leads to a solution by calculus!