Question

What is the width of a rectangular room with an area of 72 square feet and a length that is 1 foot more than three times the width?

Answers

GPT-4o mini
To find the width of the rectangular room, we'll denote the width by \( w \) (in feet). According to the problem, the length \( l \) can be expressed in terms of the width as follows:

\[
l = 3w + 1
\]

We also know that the area of the rectangle is given by the formula:

\[
\text{Area} = l \times w
\]

We are given that the area is 72 square feet, so we can write:

\[
l \times w = 72
\]

Substituting the expression for the length \( l \) into the area equation gives:

\[
(3w + 1) \times w = 72
\]

Expanding this equation:

\[
3w^2 + w = 72
\]

Rearranging it into standard quadratic form, we get:

\[
3w^2 + w - 72 = 0
\]

Now, we will solve this quadratic equation using the quadratic formula:

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

Here, \( a = 3 \), \( b = 1 \), and \( c = -72 \). First, we calculate the discriminant:

\[
b^2 - 4ac = 1^2 - 4 \times 3 \times (-72)
\]
\[
= 1 + 864
\]
\[
= 865
\]

Now we can calculate \( w \):

\[
w = \frac{-1 \pm \sqrt{865}}{2 \times 3}
\]

Calculating \( \sqrt{865} \):

\[
\sqrt{865} \approx 29.4
\]

Thus,

\[
w = \frac{-1 \pm 29.4}{6}
\]

We will only consider the positive solution for width:

\[
w = \frac{-1 + 29.4}{6} = \frac{28.4}{6} \approx 4.7333
\]

To express this more precisely, we can compute:

\[
w \approx 4.7333 \text{ feet}
\]

Finally, we can find the length:

\[
l = 3w + 1 \approx 3(4.7333) + 1 \approx 14.2 + 1 \approx 15.2 \text{ feet}
\]

To confirm, let's check the area:

\[
\text{Area} = l \times w \approx 15.2 \times 4.7333 = 72
\]

The width of the rectangular room is therefore approximately:

\[
\boxed{4.73} \text{ feet}
\]

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