To solve this problem, let's break down the steps to find the value of $x$ that maximizes the volume of the box created.

Step-by-Step Solution

1. Define Dimensions of the Original Sheet:

   • The sheet is $2$ inches by $6$ inches.

2. Define the Dimensions of the Box After Folding:

   • When squares of side length $x$ are cut from each corner, the length, width, and height of the resulting box change:
     • Length: $6 - 2x$
     • Width: $2 - 2x$
     • Height: $x$ (since $x$ is the height after folding the edges up)

3. Express the Volume of the Box as a Function of $x$:

   • The volume $V$ of the box can be expressed as: $V(x) = (6 - 2x)(2 - 2x) \cdot x$
   • Expanding this expression: $V(x) = (12 - 12x + 4x^2) \cdot x$ $V(x) = 4x^3 - 12x^2 + 12x$

4. Find the Value of $x$ that Maximizes $V(x)$:

   • To find the maximum volume, we need to find the critical points by taking the derivative of $V(x)$ with respect to $x$ and setting it to zero. $V'(x) = 12x^2 - 24x + 12$
   • Setting $V'(x) = 0$ to find critical points: $12x^2 - 24x + 12 = 0$
   • Simplifying: $x^2 - 2x + 1 = 0$ $(x - 1)^2 = 0$ $x = 1$

5. Verify the Value of $x$ and Calculate the Maximum Volume:

   • Substitute $x = 1$ back into the volume function: $V(1) = 4(1)^3 - 12(1)^2 + 12(1)$ $V(1) = 4 - 12 + 12 = 4$

Final Answer

• The value of $x$ that maximizes the volume is 1 inch.
• The maximum volume of the box is 4 cubic inches.

Would you like a more detailed breakdown of any part of the solution?

Related Questions

1. How would the volume formula change if the original sheet dimensions were different?
2. What would happen to the volume if the side length $x$ is larger than 1 inch?
3. How can we verify that $x = 1$ gives a maximum volume rather than a minimum?
4. What is the impact on volume if the sheet was square rather than rectangular?
5. Can this problem be solved using a different optimization technique?

Tip

For problems involving optimization, remember to always check the critical points by setting the derivative to zero and confirm if they yield a maximum or minimum using the second derivative or by analyzing the function's behavior.