Question

A square in sheet of side 12 inches is converted into a box with open top in the following steps. The sheet is placed horizontally. Then, equal-sized squares, each of side $x$ inches, are cut from the four corners of the sheet. Finally, the four resulting sides are bent vertically upwards in the shape of a box. If $x$ is an integer, then what value of $x$ maximizes the volume of the box?

• A. 3
• B. 2.4
• C. 3.1
• D. 4.2

Hint:

Use optimization techniques such as differentiation to maximize or minimize quantities like volume in geometric problems.

Answer 1

The Correct Option is C

The volume $V$ of the box formed is given by the formula: \[ V(x) = x(12 - 2x)^2 \] To maximize the volume, we differentiate $V(x)$ with respect to $x$ and set it equal to zero to find the critical points. After solving, we find that the value of $x$ that maximizes the volume is $3.1$.