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:
\[
\boxed{4.73} \text{ feet}
\]
\[
l = 3w + 1
\]
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:
\[
\boxed{4.73} \text{ feet}
\]