Question

A rectangular front porch has a perimeter of 84 feet. Its area is 392 square feet. What are the dimensions of the porch?

feet by
feet

Answer 1

Let the length of the porch be \( l \) feet and the width be \( w \) feet. We have two equations based on the given information:

1. The perimeter of the porch is given by:
\[
2l + 2w = 84
\]
Dividing the entire equation by 2, we can simplify it to:
\[
l + w = 42
\]

2. The area of the porch is given by:
\[
lw = 392
\]

Now, we can express \( w \) in terms of \( l \) from the first equation:
\[
w = 42 - l
\]

Next, we substitute \( w \) into the area equation:
\[
l(42 - l) = 392
\]
This expands to:
\[
42l - l^2 = 392
\]
Rearranging it gives us a quadratic equation:
\[
l^2 - 42l + 392 = 0
\]

We can solve this quadratic equation using the quadratic formula, where \( a = 1 \), \( b = -42 \), and \( c = 392 \):
\[
l = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Calculating the discriminant:
\[
b^2 - 4ac = (-42)^2 - 4(1)(392) = 1764 - 1568 = 196
\]
Now, substituting the values into the formula:
\[
l = \frac{42 \pm \sqrt{196}}{2(1)} = \frac{42 \pm 14}{2}
\]
This results in two possible solutions for \( l \):
1.
\[
l = \frac{42 + 14}{2} = \frac{56}{2} = 28
\]
2.
\[
l = \frac{42 - 14}{2} = \frac{28}{2} = 14
\]

Now, we find the corresponding widths \( w \):
- If \( l = 28 \):
\[
w = 42 - 28 = 14
\]
- If \( l = 14 \):
\[
w = 42 - 14 = 28
\]

Thus, the dimensions of the porch are:
\[
\text{Dimensions: } 28 \text{ feet by } 14 \text{ feet}
\]

In conclusion, the dimensions of the front porch are **28 feet by 14 feet**.