Sure, let's solve the equation [tex]\( V = \sqrt[3]{\frac{a x^2 h}{b - h}} \)[/tex] for [tex]\( h \)[/tex]. Follow these steps:
1. Given the equation:
[tex]\[
V = \sqrt[3]{\frac{a x^2 h}{b - h}}
\][/tex]
2. To eliminate the cube root, cube both sides of the equation:
[tex]\[
V^3 = \left( \sqrt[3]{\frac{a x^2 h}{b - h}} \right)^3
\][/tex]
[tex]\[
V^3 = \frac{a x^2 h}{b - h}
\][/tex]
3. Now, to isolate [tex]\( h \)[/tex] on one side, multiply both sides of the equation by [tex]\( b - h \)[/tex]:
[tex]\[
V^3 (b - h) = a x^2 h
\][/tex]
4. Distribute [tex]\( V^3 \)[/tex] on the left-hand side:
[tex]\[
V^3 b - V^3 h = a x^2 h
\][/tex]
5. Gather all terms involving [tex]\( h \)[/tex] on one side of the equation:
[tex]\[
V^3 b = a x^2 h + V^3 h
\][/tex]
6. Factor out [tex]\( h \)[/tex] from the terms on the right-hand side:
[tex]\[
V^3 b = h (a x^2 + V^3)
\][/tex]
7. Finally, solve for [tex]\( h \)[/tex] by dividing both sides by [tex]\( a x^2 + V^3 \)[/tex]:
[tex]\[
h = \frac{V^3 b}{a x^2 + V^3}
\][/tex]
Hence, the value of [tex]\( h \)[/tex] expressed in terms of [tex]\( V \)[/tex], [tex]\( a \)[/tex], [tex]\( x \)[/tex], and [tex]\( b \)[/tex] is:
[tex]\[
h = \frac{V^3 b}{a x^2 + V^3}
\][/tex]
