Questions: A rectangle has a length of 16 units and a width of 4 units. Squares of x by x units are cut out of each corner, and then the sides are folded up to create an open box. Express the volume of the box as a polynomial function in terms of x.

A rectangle has a length of 16 units and a width of 4 units. Squares of x by x units are cut out of each corner, and then the sides are folded up to create an open box.

Express the volume of the box as a polynomial function in terms of x.

Transcript text: A rectangle has a length of 16 units and a width of 4 units. Squares of $x$ by $x$ units are cut out of each corner, and then the sides are folded up to create an open box. Express the volume of the box as a polynomial function in terms of $x$.

Solution

Solution Steps

Step 1: Understanding the Problem

Given a rectangle with length $L$ units and width $W$ units, squares of side length $x$ units are cut from each corner. The sides are then folded up to create an open box. We need to express the volume of this box as a polynomial function of $x$.

Step 2: Deriving the Volume Formula

After cutting out the squares, the new dimensions of the base of the box are $L - 2x$ by $W - 2x$. The height of the box is $x$, as this is the dimension of the folded part. Therefore, the volume $V(x)$ can be expressed as: \[V(x) = (L - 2x)(W - 2x)x\]

Step 3: Expanding the Volume Formula

Expanding the formula, we get: \[V(x) = LWx - 2Lx^2 - 2Wx^2 + 4x^3\] Simplifying, we obtain: \[V(x) = 4x^3 - 2(L + W)x^2 + LWx\]

Step 4: Calculating the Volume

Substituting $L = 16$, $W = 4$, and $x = 1$ into the volume formula, we calculate the volume as $V(x) = 28$ units^3, rounded to 2 decimal places.