Sure, let's break down the given problem step-by-step.
### Given Information:
1. The length of the rectangular prism (box) is given as 24 inches.
2. The height of the box is represented as [tex]\( x \)[/tex] inches.
3. The width of the box is 7 inches less than the height, which makes the width equal to [tex]\( x - 7 \)[/tex] inches.
### Volume of a Rectangular Prism:
The volume [tex]\( V \)[/tex] of a rectangular prism is given by the formula:
[tex]\[ V = \text{length} \times \text{width} \times \text{height} \][/tex]
Plugging in the given values:
[tex]\[ V = 24 \times (x - 7) \times x \][/tex]
### Simplifying the Volume Expression:
First, distribute [tex]\( x \)[/tex] across the terms inside the parenthesis:
[tex]\[ V = 24 \times (x^2 - 7x) \][/tex]
Then, distribute 24 inside the parenthesis:
[tex]\[ V = 24x^2 - 168x \][/tex]
So, the equation that models the volume of the box in terms of its height [tex]\( x \)[/tex] is:
[tex]\[ V = 24x^2 - 168x \][/tex]
### Can the Height of the Box be 15 inches?
To determine whether the height of the box can be 15 inches, we need to plug [tex]\( x = 15 \)[/tex] into the simplified volume expression and verify a few things:
1. Calculate the width when [tex]\( x = 15 \)[/tex]:
[tex]\[ \text{Width} = 15 - 7 = 8 \text{ inches} \][/tex]
2. Calculate the volume when [tex]\( x = 15 \)[/tex]:
[tex]\[ V = 24 \times 15^2 - 168 \times 15 \][/tex]
3. Ensure that the volume is non-negative (since volume cannot be negative):
Let's simplify this step further:
[tex]\[ 24 \times 15^2 - 168 \times 15 = 24 \times 225 - 2520 \][/tex]
[tex]\[ = 5400 - 2520 = 2880 \][/tex]
Since [tex]\( 2880 \)[/tex] is a positive value, the calculated volume is non-negative. This verifies that having a height of 15 inches results in a valid volume.
Thus, it is viable for the height of the box to be 15 inches.
### Summary:
1. Volume Formula Modeling the Box in terms of Height [tex]\( x \)[/tex]: [tex]\( 24x^2 - 168x \)[/tex]
2. Width when Height is 15 Inches: 8 inches
3. Viable for Height to be 15 Inches: Yes
Therefore:
- The equation modeling the volume of the box in terms of its height [tex]\( x \)[/tex] is [tex]\( V = 24x^2 - 168x \)[/tex].
- The width is 8 inches when the height is 15 inches.
- The height can be 15 inches since it results in a non-negative volume value.
