Answer:
Required,
[tex]L=\frac{3+\sqrt{37}}{2}-\frac{144}{x-3}[/tex]
[tex]W=\frac{3-\sqrt{37}}{2}-\frac{144}{x-3}[/tex]
Step-by-step explanation:
Given volume and depth respectively,
[tex]V(x)=x^3+13x-210[/tex] and [tex]x-3[/tex]
To find length and width of the rectanglular swiming pool we know,
Volume=length[tex]\times[/tex]height[tex]\times[/tex]depth.
Let depth=D=x-3, length=L, width=W, then
[tex]V=DLW[/tex]
[tex]x^3+13x-210= LW(x-3)[/tex]
[tex]LW=\frac{x^3+13x-210}{x-3}[/tex]
After divide we will get [tex]x^2-3x-22[/tex] with remainder -144.
Thus,
[tex]x^3+13x-210=(x^2-3x-22)(x-3)-144=(x-3)LW[/tex]
Now to find root of,
[tex]x^2-3x-144=\frac{3\pm\sqrt{9+88}}{2}=\frac{3\pm \sqrt{37}}{2}[/tex]
Thus,
[tex]L=\frac{3+\sqrt{37}}{2}-\frac{144}{x-3}[/tex]
[tex]W=\frac{3-\sqrt{37}}{2}-\frac{144}{x-3}[/tex]
