To find the width of the deck space around the hot tub, [tex]$x$[/tex], we will follow the steps provided:
### Step 1:
Write an equation for the area of the enclosed space.
Since the hot tub is square with a side length of 6 feet and it is surrounded by [tex]$x$[/tex] feet of deck on each side, the side length of the entire enclosed space (including the deck) will be [tex]\(6 + 2x\)[/tex]. Therefore, the area, [tex]\(y\)[/tex], of the enclosed space is given by:
[tex]\[
y = (6 + 2x)^2
\][/tex]
### Step 2:
Substitute the given area of the enclosed space into the equation.
We are given that the area of the enclosed space is 169 square feet. So, we substitute 169 for [tex]\(y\)[/tex] in our equation:
[tex]\[
169 = (6 + 2x)^2
\][/tex]
### Step 3:
Solve the equation for [tex]\(x\)[/tex].
First, take the square root of both sides to eliminate the square.
[tex]\[
\sqrt{169} = 6 + 2x
\][/tex]
Since [tex]\(\sqrt{169}\)[/tex] is 13, we have:
[tex]\[
13 = 6 + 2x
\][/tex]
Next, solve for [tex]\(x\)[/tex] by isolating it on one side of the equation. Subtract 6 from both sides:
[tex]\[
13 - 6 = 2x
\][/tex]
This simplifies to:
[tex]\[
7 = 2x
\][/tex]
Finally, divide both sides by 2 to solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{7}{2} = 3.5
\][/tex]
So, the width of the deck space around the hot tub, [tex]\(x\)[/tex], is 3.5 feet.
To summarize, the side length of the entire enclosed space is 13 feet, and the width of the deck space around the hot tub is 3.5 feet.
