We have to find the expression for the volume of the box in terms of its height (x).
Then, the height is h = x
The length is l = 24 in.
The width is w = x - 7 in, as it is 7 inches less than the height.
The volume is 2880 cubic inches.
We then can express the volume as:
[tex]\begin{gathered} V=2880 \\ l\cdot w\cdot h=2880 \\ 24\cdot(x-7)\cdot x=2880 \\ 24(x^2-7x)=2880 \\ 24x^2-168x=2880 \end{gathered}[/tex]Then, the blancks are filled with 24, 168 and 2880.
We now have to check if the height of the box can be 15 inches.
We can replace x with 15 and see if the equation is still valid:
[tex]\begin{gathered} 24(15)^2-168(15)=2880 \\ 24\cdot225-2520=2880 \\ 5400-2520=2880 \\ 2880=2880\longrightarrow\text{True} \end{gathered}[/tex]It is possible that the height is 15 in.
Answer:
The volume of the box is 24x² - 168x = 2880.
Yes, it is possible that the height is 15 inches.