Good morning.

Quick question.

dr/dt=K/r^2

When the radius is 3cm, the radius is increasing at a rate of 5/36pi. Use this fact to find the constant K.

I know that the general form of these growth factors is e^kt but im not sure where the rate and the original radius goes.

Is it e^5/36pi*k=3? doesnt seem right.

Hang on ive just realized I need to use the equation for the volume of a sphere first right? v=4/3pir^3 to find a value to plug into my previous equation. Am I on the right track?

yes indeed

Good morning!

To find the constant K in the given differential equation, we can use the fact that when the radius is 3 cm, the rate of change (dr/dt) is 5/36π.

First, let's rewrite the given differential equation:
dr/dt = K/r^2

Now, we can substitute the given values into the equation:
5/36π = K/(3^2)

Next, let's simplify the equation:
5/36π = K/9

To isolate K, we can cross-multiply:
5 * 9 = 36π * K
45 = 36π * K

Finally, divide both sides of the equation by 36π to solve for K:
K = 45 / (36π)

So, the constant K in the differential equation is K = 45 / (36π).

I hope this helps! Let me know if you have any further questions.