An open-top box with a square base is to have a volume of 4 cubic feet. Find
the dimensions of the box (in ft) that can be made with the smallest amount of material

To find the dimensions of the open-top box with the smallest amount of material, we need to minimize the surface area of the box while keeping its volume constant at 4 cubic feet.

Let's denote:
- \( x \) as the side length of the square base,
- \( h \) as the height of the box.

Given:
\[ x^2h = 4 \]

The surface area \( S \) of the box (with an open top) is given by:
\[ S = x^2 + 4xh \]

First, express \( h \) in terms of \( x \) using the volume constraint:
\[ h = \frac{4}{x^2} \]

Substitute this into the surface area formula:
\[ S = x^2 + 4x \left( \frac{4}{x^2} \right) \]
\[ S = x^2 + \frac{16}{x} \]

To minimize \( S \), we take the derivative of \( S \) with respect to \( x \) and set it to zero:
\[ \frac{dS}{dx} = 2x - \frac{16}{x^2} \]

Set the derivative equal to zero:
\[ 2x - \frac{16}{x^2} = 0 \]

Solve for \( x \):
\[ 2x = \frac{16}{x^2} \]
\[ 2x^3 = 16 \]
\[ x^3 = 8 \]
\[ x = 2 \]

Now, find the corresponding height \( h \):
\[ h = \frac{4}{x^2} = \frac{4}{2^2} = 1 \]

Thus, the dimensions of the box that use the smallest amount of material are:
- Side length of the square base: \( x = 2 \) feet
- Height of the box: \( h = 1 \) foot

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

Here are 5 related questions you can ask:
1. How do you find the critical points for minimizing or maximizing a function?
2. Can you explain why the second derivative test is useful in this context?
3. How would the problem change if the box had a closed top?
4. What are the applications of optimization in real-life scenarios?
5. Can you provide another example of an optimization problem?

**Tip:** When solving optimization problems, always check the second derivative to ensure that you have found a minimum (or maximum) point.

Learn how to find the dimensions of an open-top box with the smallest amount of material usage, maintaining a constant volume of 4 cubic feet. This problem involves optimizing the surface area of the box, balancing constraints and formulas related to volume and surface area.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation