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Gabriel is designing equally sized horse stalls that are each in the shape of a rectangular prism. Each stall must be 9 feet high and have a volume of 1,080 cubic feet. The length of each stall should be 2 feet longer than its width.

The volume of a rectangular prism is found using the formula [tex]$V = l \cdot w \cdot h$[/tex], where [tex]$l$[/tex] is the length, [tex][tex]$w$[/tex][/tex] is the width, and [tex]$h$[/tex] is the height.

Complete the equation that represents the volume of a stall in terms of its width of [tex]$x$[/tex] feet.

[tex]x^2 + \square[/tex]
[tex]\square x = \square[/tex]

Is it possible for the width of a stall to be 10 feet? [tex]\square[/tex]



Answer :

GinnyAnswer
We start with the given information:
- Height [tex]\(h = 9\)[/tex] feet.
- Volume [tex]\(V = 1080\)[/tex] cubic feet.
- The length [tex]\(l\)[/tex] is 2 feet longer than the width [tex]\(w\)[/tex].

We need to determine if it's possible for the width of a stall to be 10 feet. To do this, we first formulate an equation representing the volume of the stall in terms of its width [tex]\(x\)[/tex].

### Step-by-Step Solution:

1. Express the length in terms of the width:
Since the length [tex]\(l\)[/tex] is 2 feet longer than the width [tex]\(w\)[/tex], let:
[tex]\[ l = x + 2 \][/tex]

2. Write the volume formula using the given dimensions:
The volume of the stall [tex]\(V\)[/tex] is given by the formula for the volume of a rectangular prism:
[tex]\[ V = l \cdot w \cdot h \][/tex]

3. Substitute the given values and the expression for [tex]\(l\)[/tex]:
[tex]\[ 1080 = (x + 2) \cdot x \cdot 9 \][/tex]

4. Simplify the equation:
[tex]\[ 1080 = 9 \cdot x \cdot (x + 2) \][/tex]

5. Distribute and combine like terms:
[tex]\[ 1080 = 9x^2 + 18x \][/tex]

Thus, the complete equation representing the volume of a stall in terms of its width [tex]\(x\)[/tex] is:
[tex]\[ 9x^2 + 18x = 1080 \][/tex]

To check if it is possible for the width [tex]\(x\)[/tex] to be 10 feet:
1. Substitute [tex]\(x = 10\)[/tex] into the equation:
[tex]\[ 9 \cdot 10^2 + 18 \cdot 10 = 1080 \][/tex]

2. Calculate the left-hand side:
[tex]\[ 9 \cdot 100 + 180 = 1080 \][/tex]
[tex]\[ 900 + 180 = 1080 \][/tex]

Since the equation holds true, it confirms that it is possible for the width of a stall to be 10 feet.

### Summary

Fill the drop-down menus accordingly:
1. [tex]\(9x^2 + 18x = 1080\)[/tex]
2. It is possible for the width of a stall to be 10 feet.

Therefore:
[tex]$x^2+$[/tex] 9
18 [tex]$x=$[/tex] 1080
Is it possible for the width of a stall to be 10 feet? True