Question

John was told that the rectangle had a side length of 2x+12 find the other side length was 3x+30.  If the perimeter of the rectangle was 164 ft then what is the value of x ?

Answer 1

To find the value of \( x \) given the side lengths of the rectangle and the perimeter, we can use the formula for the perimeter of a rectangle:

\[
P = 2(l + w)
\]

where \( l \) is the length and \( w \) is the width. Here, the length and width are given as \( 2x + 12 \) and \( 3x + 30 \), respectively, and the perimeter is given as 164 ft.

First, we set up the equation for the perimeter:

\[
2((2x + 12) + (3x + 30)) = 164
\]

Now, simplify the equation:

\[
2(2x + 12 + 3x + 30) = 164
\]

Combine like terms inside the parentheses:

\[
2(5x + 42) = 164
\]

Now, divide both sides by 2:

\[
5x + 42 = 82
\]

Next, isolate \( 5x \) by subtracting 42 from both sides:

\[
5x = 82 - 42
\]
\[
5x = 40
\]

Now, divide by 5 to find \( x \):

\[
x = \frac{40}{5} = 8
\]

Thus, the value of \( x \) is:

\[
\boxed{8}
\]

To verify, we can calculate the side lengths and the perimeter:

1. For \( x = 8 \):
- Length: \( 2(8) + 12 = 16 + 12 = 28 \)
- Width: \( 3(8) + 30 = 24 + 30 = 54 \)

2. Now, check the perimeter:
\[
P = 2(28 + 54) = 2(82) = 164
\]

Since the computed perimeter matches the given perimeter, our solution is correct. The value of \( x \) is indeed \( \boxed{8} \).