Sure, let's go through the problem systematically.
We have a box with the following properties:
- The length [tex]\( l \)[/tex] is 24 inches.
- The volume [tex]\( V \)[/tex] is 2,880 cubic inches.
- The width [tex]\( w \)[/tex] is 7 inches less than the height [tex]\( h \)[/tex].
First, express the width [tex]\( w \)[/tex] in terms of the height [tex]\( h \)[/tex]:
[tex]\[ w = h - 7 \][/tex]
Next, use the volume formula for a rectangular prism:
[tex]\[ V = l \times w \times h \][/tex]
Substitute the given values and the expression for the width:
[tex]\[ 2,880 = 24 \times (h - 7) \times h \][/tex]
This simplifies to:
[tex]\[ 2,880 = 24h(h - 7) \][/tex]
Divide both sides by 24:
[tex]\[ 120 = h(h - 7) \][/tex]
So, the equation is:
[tex]\[ h^2 - 7h - 120 = 0 \][/tex]
Now, we should solve this quadratic equation. Factoring or using the quadratic formula, we get the solutions:
[tex]\[ h^2 - 7h - 120 = 0 \][/tex]
[tex]\[ h = 15 \quad \text{or} \quad h = -8 \][/tex]
Since a height of -8 inches is not physically meaningful, we discard it. Therefore, the only practical solution is:
[tex]\[ h = 15 \][/tex]
So, the height of the box can indeed be 15 inches.
We can now summarize the correct answers to the question:
1. Complete the equation:
[tex]\[
(x \quad) \quad x^2-7x = 2880
\][/tex]
2. Is it possible for the height of the box to be 15 inches?
[tex]\[
Yes
\][/tex]
