The height and length of the box to the nearest tenth will be 10.6 inch and 7.4 inch respectively.
Explanation
Suppose, the height of the box is [tex]h[/tex] inch
As the length of the box is 3.2 in. less than the height, so the length will be: [tex](h-3.2) inch[/tex]
Given that, the width of the box is 2.3 inch. and the volume is 180.4 inch³
Formula for Volume of a box: [tex]V= length*width*height[/tex]
So, the equation will be....
[tex](h-3.2)*2.3*h = 180.4\\ \\ 2.3h^2 -7.36h=180.4\\ \\ 2.3h^2-7.36h-180.4=0[/tex]
Now using quadratic formula......
[tex]h=\frac{-b+/-\sqrt{b^2-4ac}}{2a}\\ \\ h= \frac{-(-7.36)+/-\sqrt{(-7.36)^2-4(2.3)(-180.4)}}{2(2.3)}\\ \\ h= \frac{7.36+/-\sqrt{1713.8496}}{4.6} \\ \\ h=\frac{7.36+/-41.398...}{4.6} \\ \\ h= 10.599... or -7.399...[/tex]
(Negative value of [tex]h[/tex] is ignored as the height can't be negative)
So, the height to the nearest tenth will be 10.6 inch and the length will be: [tex](10.6-3.2)inch= 7.4 inch[/tex]
