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We are tasked with determining the size of the squares to cut from the corners of a 10-inch by 16-inch piece of cardboard, which will allow us to create a rectangular box with an open top and the largest possible volume.

### Let's set up the problem:

1. The cardboard is **10 inches by 16 inches**.
2. Small squares of side length **x** inches are cut from each corner.
3. After cutting and folding up the sides, the dimensions of the base of the box will be:
   - **Length**: \( 16 - 2x \) inches
   - **Width**: \( 10 - 2x \) inches
4. The **height** of the box will be \( x \) inches (the size of the square cut and folded).

Thus, the volume **V** of the box is given by:
\[
V(x) = x(16 - 2x)(10 - 2x)
\]
We need to maximize this volume.

### Steps to solve:
1. Expand the volume function.
2. Find the critical points by taking the derivative of the volume function and setting it equal to zero.
3. Determine the value of **x** that maximizes the volume by solving the equation and confirming it with the second derivative test.

Let me compute the optimal **x** for maximum volume.The critical points for the volume function are **x = 2** and **x = 20/3** (which is approximately 6.67). However, since the box dimensions must be positive, \( x = 20/3 \) would result in negative base dimensions, making it an invalid solution.

Thus, the valid solution is \( x = 2 \).

This means the squares cut from each corner should have a side length of **2 inches** to maximize the volume of the box.

Would you like further details on this calculation, or perhaps help with related questions?

### Related questions:
1. What is the maximum volume achieved with \( x = 2 \)?
2. What happens to the volume if a different size of square is cut?
3. Can you solve this using a different approach, like trial and error?
4. How does the second derivative confirm that this is a maximum?
5. What would the dimensions of the box be after cutting?

#### Tip:
In optimization problems, always check boundary conditions and critical points to ensure you're maximizing or minimizing correctly.

A rectangular box with an open top is to be made from a 10in x 16in piece of cardboard by removing small squares of equal size from the corners and folding up the remaining sides. What should be the size of the squares cut from the corners so that the box will have the largest possible volume? (Disregard the thickness of the cardboard.)

Learn how to find the optimal size of squares to cut from the corners of a 10-inch by 16-inch cardboard to maximize the volume of an open-top rectangular box. This solution involves optimization using algebraic techniques and first derivative tests. Ideal for students in Grades 9-12.

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