Let's break down the problem:
1. Volume formula:
The formula for the volume of a rectangular prism is given by:
[tex]\[
V = l \cdot w \cdot h
\][/tex]
Here, 'V' is the volume, 'l' is the length, 'w' is the width, and 'h' is the height of the rectangular prism.
2. Given information:
- Height ([tex]\( h \)[/tex]) = 9 feet
- Volume ([tex]\( V \)[/tex]) = 1,080 cubic feet
- Length ([tex]\( l \)[/tex]) is 2 feet longer than the width ([tex]\( w \)[/tex]). Therefore, [tex]\( l = w + 2 \)[/tex].
3. Substitute given values and relationships into the volume formula:
[tex]\[
1080 = (w + 2) \cdot w \cdot 9
\][/tex]
4. Simplify the equation:
Divide both sides of the equation by 9 to isolate the width terms:
[tex]\[
\frac{1080}{9} = (w + 2) \cdot w
\][/tex]
[tex]\[
120 = w^2 + 2w
\][/tex]
5. Rewrite the equation in standard quadratic form:
[tex]\[
w^2 + 2w - 120 = 0
\][/tex]
So, the equation that represents the volume of a stall in terms of its width [tex]\( w \)[/tex] is:
[tex]\[
w^2 + 2w - 120 = 0
\][/tex]
6. Determine the width of the stall:
To solve for [tex]\( w \)[/tex], we need to find the roots of the quadratic equation [tex]\( w^2 + 2w - 120 = 0 \)[/tex].
The solutions to this quadratic equation are:
[tex]\[
w = -12 \quad \text{and} \quad w = 10
\][/tex]
However, since width cannot be negative, we discard [tex]\( w = -12 \)[/tex], leaving us with:
[tex]\[
w = 10
\][/tex]
So, after evaluating the solution of the quadratic equation, it is possible for the width of a stall to be 10 feet.
To summarize:
- The equation that represents the volume of a stall in terms of its width [tex]\( w \)[/tex] is:
[tex]\[
w^2 + 2w - 120 = 0
\][/tex]
- Is it possible for the width of a stall to be 10 feet? Yes.
