To solve the problem step-by-step, we will use the information provided about the horse stalls:
1. Given Data:
- Height of the stall (h) = 9 feet
- Volume of the stall (V) = 1080 cubic feet
- Length of the stall (l) = width (w) + 2 feet
2. Volume Formula:
The volume of a rectangular prism is calculated using the formula:
[tex]\[
V = l \cdot w \cdot h
\][/tex]
3. Substitute the Given Information:
Let's denote the width of the stall by [tex]\( x \)[/tex] feet. Therefore, the length will be [tex]\( x + 2 \)[/tex] feet.
Now substitute [tex]\( l = x + 2 \)[/tex], [tex]\( w = x \)[/tex], and [tex]\( h = 9 \)[/tex] into the volume formula:
[tex]\[
1080 = (x + 2) \cdot x \cdot 9
\][/tex]
4. Simplify the Equation:
Divide both sides by 9 to simplify:
[tex]\[
120 = (x + 2) \cdot x
\][/tex]
6. Form a Quadratic Equation:
Expand and rearrange the equation to set it to zero:
[tex]\[
120 = x^2 + 2x \implies x^2 + 2x - 120 = 0
\][/tex]
7. Solve the Quadratic Equation:
Solve the quadratic equation [tex]\( x^2 + 2x - 120 = 0 \)[/tex] to find the possible values for [tex]\( x \)[/tex]. The solutions to this equation are:
[tex]\[
x = -12 \quad \text{or} \quad x = 10
\][/tex]
Since the width of a stall cannot be negative, the only feasible solution is [tex]\( x = 10 \)[/tex].
8. Determine Feasibility:
We check if 10 feet is a possible width.
To answer the drop-down questions:
Complete the equation that represents the volume of a stall in terms of its width of [tex]\( x \)[/tex] feet.
The equation is:
[tex]\[
\boxed{x^2 + 2x - 120 = 0}
\][/tex]
Is it possible for the width of a stall to be 10 feet?
Yes, it is possible. The width of the stall can be 10 feet.
[tex]\[
\boxed{\text{Yes}}
\][/tex]
Therefore:
- The correct form of the equation representing the volume of a stall in terms of width [tex]\( x \)[/tex]:
[tex]\[
x^2 + 2x - 120 = 0
\][/tex]
- It is indeed possible for the width of a stall to be 10 feet.
