Question

The volume of a rectangular prism is 216 cubic feet. Its height is 6 feet, and its length is 1 foot shorter than 2 times its width.
How long and wide is the prism?
feet long and
feet wide

Answer 1

To find the dimensions of the rectangular prism, we can start by using the formula for the volume of a rectangular prism, which is:

\[
\text{Volume} = \text{length} \times \text{width} \times \text{height}
\]

Given:
- Volume = 216 cubic feet
- Height = 6 feet
- Length = \(2 \times \text{width} - 1\)

Let's denote the width as \(w\). Then, we can express the length as:

\[
l = 2w - 1
\]

Substituting \(l\) and the height into the volume formula gives us:

\[
216 = (2w - 1) \times w \times 6
\]

Now simplify the equation:

\[
216 = 6(2w - 1)w
\]

Divide both sides by 6:

\[
36 = (2w - 1)w
\]

Expanding the right side:

\[
36 = 2w^2 - w
\]

Rearranging gives:

\[
2w^2 - w - 36 = 0
\]

Now we can solve this quadratic equation using the quadratic formula:

\[
w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

where \(a = 2\), \(b = -1\), and \(c = -36\).

Calculating the discriminant:

\[
b^2 - 4ac = (-1)^2 - 4 \cdot 2 \cdot (-36) = 1 + 288 = 289
\]

Now apply the quadratic formula:

\[
w = \frac{-(-1) \pm \sqrt{289}}{2 \cdot 2} = \frac{1 \pm 17}{4}
\]

This gives us two possible solutions for \(w\):

1. \(w = \frac{1 + 17}{4} = \frac{18}{4} = 4.5\)
2. \(w = \frac{1 - 17}{4} = \frac{-16}{4} = -4\) (not a valid solution since width cannot be negative)

Thus, the width is:

\[
w = 4.5 \text{ feet}
\]

Now we can find the length using the expression for length:

\[
l = 2w - 1 = 2(4.5) - 1 = 9 - 1 = 8 \text{ feet}
\]

So, the dimensions of the prism are:

\[
\text{Length: } 8 \text{ feet}
\]
\[
\text{Width: } 4.5 \text{ feet}
\]

Therefore, the rectangular prism is **8 feet long and 4.5 feet wide**.