Certainly! Let's work on making [tex]\( h \)[/tex] the subject of the equation:
[tex]\[
V = \sqrt[3]{\frac{9 x^2 h}{b - h}}
\][/tex]
Here is the step-by-step solution:
1. Start with the given equation:
[tex]\[
V = \sqrt[3]{\frac{9 x^2 h}{b - h}}
\][/tex]
2. To eliminate the cube root, cube both sides of the equation:
[tex]\[
V^3 = \left(\sqrt[3]{\frac{9 x^2 h}{b - h}}\right)^3
\][/tex]
This simplifies to:
[tex]\[
V^3 = \frac{9 x^2 h}{b - h}
\][/tex]
3. Next, isolate the term involving [tex]\( h \)[/tex] on one side of the equation by multiplying both sides by [tex]\((b - h)\)[/tex]:
[tex]\[
V^3 (b - h) = 9 x^2 h
\][/tex]
4. Distribute [tex]\( V^3 \)[/tex] on the left side:
[tex]\[
V^3 b - V^3 h = 9 x^2 h
\][/tex]
5. To isolate [tex]\( h \)[/tex], get all [tex]\( h \)[/tex]-terms on one side of the equation. Add [tex]\( V^3 h \)[/tex] to both sides:
[tex]\[
V^3 b = 9 x^2 h + V^3 h
\][/tex]
6. Factor [tex]\( h \)[/tex] out of the right side of the equation:
[tex]\[
V^3 b = h (9 x^2 + V^3)
\][/tex]
7. Finally, solve for [tex]\( h \)[/tex] by dividing both sides by [tex]\( (9 x^2 + V^3) \)[/tex]:
[tex]\[
h = \frac{V^3 b}{9 x^2 + V^3}
\][/tex]
Therefore, the expression for [tex]\( h \)[/tex] in terms of [tex]\( x \)[/tex], [tex]\( b \)[/tex], and [tex]\( V \)[/tex] is:
[tex]\[
h = \frac{1.0 \times 10^{39} V^3 b}{1.0 \times 10^{39} V^3 + 8.99999999999995 \times 10^{39} x^2}
\][/tex]
