rearrange for dh/dt 16*π*​*h*2*dtdh*​=3−*kh*3

To rearrange the equation for \(\frac{dh}{dt}\), let's start with the given equation:

\[
16\pi h^2 \frac{dt}{dh} = 3 - kh^3
\]

### Step 1: Isolate \(\frac{dt}{dh}\)
First, solve for \(\frac{dt}{dh}\) by dividing both sides of the equation by \(16\pi h^2\):

\[
\frac{dt}{dh} = \frac{3 - kh^3}{16\pi h^2}
\]

### Step 2: Invert to find \(\frac{dh}{dt}\)
To find \(\frac{dh}{dt}\), take the reciprocal of both sides:

\[
\frac{dh}{dt} = \frac{16\pi h^2}{3 - kh^3}
\]

This is the expression for \(\frac{dh}{dt}\).

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Would you like any further clarification on this rearrangement? Here are some related questions you might find interesting:

1. How does this equation change when \(h = 0\)?
2. What happens to \(\frac{dh}{dt}\) as \(k\) increases?
3. How would you find the critical points of \(h\) based on this equation?
4. Can this equation be integrated to find \(h(t)\)?
5. What physical scenarios might this equation represent?

**Tip:** When rearranging equations, always carefully track your variables and constants to avoid confusion in later steps.

Learn how to rearrange the differential equation 16πh² dtdh = 3 - kh³ to find dh/dt. This problem involves differential equations and the separation of variables method. Suitable for advanced high school and undergraduate students.

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