To determine the height of a shipping container in terms of its width [tex]\( x \)[/tex], we start by understanding the given information:
1. The volume [tex]\( V \)[/tex] of the shipping container is given by the polynomial expression:
[tex]\[
V = 5x^3 + 7.5x^2
\][/tex]
2. The length [tex]\( L \)[/tex] of the container is five times its width [tex]\( x \)[/tex]:
[tex]\[
L = 5x
\][/tex]
3. The volume of a rectangular prism is given by the product of its length [tex]\( L \)[/tex], width [tex]\( W \)[/tex], and height [tex]\( H \)[/tex]:
[tex]\[
V = L \cdot W \cdot H
\][/tex]
Here, [tex]\( W \)[/tex] is the width, which is [tex]\( x \)[/tex]. So, substituting into the volume formula, we have:
[tex]\[
V = (5x) \cdot x \cdot H
\][/tex]
First, substitute the given polynomial for [tex]\( V \)[/tex]:
[tex]\[
5x^3 + 7.5x^2 = 5x^2 \cdot H
\][/tex]
To solve for [tex]\( H \)[/tex], we divide both sides of the equation by [tex]\( 5x^2 \)[/tex]:
[tex]\[
H = \frac{5x^3 + 7.5x^2}{5x^2}
\][/tex]
Now, we simplify the right-hand side:
[tex]\[
H = \frac{5x^3 + 7.5x^2}{5x^2} = \frac{5x^3}{5x^2} + \frac{7.5x^2}{5x^2} = x + 1.5
\][/tex]
Thus, the height [tex]\( H \)[/tex] of the shipping container in terms of its width [tex]\( x \)[/tex] is:
[tex]\[
H = x + 1.5
\][/tex]
