To solve this problem, we'll start by setting up an equation to represent the volume of the rectangular box.
We know that the volume [tex]\( V \)[/tex] of a rectangular prism is given by:
[tex]\[ V = l \cdot w \cdot h \][/tex]
Given information:
- The volume [tex]\( V \)[/tex] of the box is 2880 cubic inches.
- The length [tex]\( l \)[/tex] of the box is 24 inches.
- The width [tex]\( w \)[/tex] of the box is 7 inches less than the height [tex]\( h \)[/tex].
Let's denote the height by [tex]\( x \)[/tex]. Therefore, the width [tex]\( w \)[/tex] can be expressed as:
[tex]\[ w = x - 7 \][/tex]
Substitute these values into the volume formula:
[tex]\[ 2880 = 24 \cdot (x - 7) \cdot x \][/tex]
Expanding and simplifying this, we get:
[tex]\[ 2880 = 24x(x - 7) \][/tex]
[tex]\[ 2880 = 24x^2 - 168x \][/tex]
[tex]\[ 120 = x^2 - 7x \][/tex]
Thus, the correct equation that models the volume is:
[tex]\[ 120 = x^2 - 7x \][/tex]
Now we need to determine if it's possible for the height of the box to be 15 inches. Let's substitute [tex]\( x = 15 \)[/tex] into the equation to see if it holds true:
[tex]\[ x^2 - 7x = 15^2 - 7 \cdot 15 \][/tex]
[tex]\[ 15^2 = 225 \][/tex]
[tex]\[ 7 \cdot 15 = 105 \][/tex]
[tex]\[ 225 - 105 = 120 \][/tex]
The left side of the equation becomes 120, which equals the right side of the original equation. Hence, the given height of 15 inches satisfies the equation.
So, the answers are as follows:
- Complete the equation that models the volume of the box in terms of its height [tex]\( x \)[/tex]:
[tex]\[
120 = x^2 - 7x
\][/tex]
- Is it possible for the height of the box to be 15 inches?
[tex]\[
\text{Yes}
\][/tex]
