Need Help Finding the Radius of this sphere

[temporaryinsanit]

This is the question:

At time t, t is greater than or equal to 0, the volume of a sphere is increasing at the rate proportional to the reciprocal of its radius. At t = 0, the radius of the sphere is 1 and at t = 5, the radius is 12. [The volume of a sphere with radius r is V = (4/3)(pi)(r)^3]

What is the radius of the sphere as a function of t?

 

[tutor_joel]

So, they say that "the volume of a sphere is increasing at the rate proportional to the reciprocal of its radius." That means...

\(\displaystyle \frac{dV}{dt}\propto\frac{1}{r}\)

in order to turn a proportion into an equation, you need a constant. In this case, it's a rate constant since we are dealing with time.

\(\displaystyle \frac{dV}{dt}=k\frac{1}{r}\)

here, k is the rate constant. So we need to solve for it. That's what the initial and final conditions are for. also...

\(\displaystyle V = \frac{4\pi}{3}r^3\)

and therefore...

\(\displaystyle dV = 4\pi r^2dr\)

Substitute into rate equation and then do separation of variables. You will have 2 definite integrals to solve, and then solve for k. After you find your rate constant, resolve the equations as indefinite integrals (don't forget the C, you need to solve for it). Then you will have r(t)

Let me know if this makes sense or if you need any further help. It's a pretty cool problem.

 

[BigGlenntheHeavy]

\(\displaystyle Given: \ \frac{dV}{dt} \ = \ \frac{k}{r}, \ V \ = \ \frac{4\pi r^{3}}{3}, \ r(0) \ = \ 1, \ and \ r(5) \ = \ 12, \ find \ r(t).\)

\(\displaystyle Then \ dV \ = \ 4\pi r^{2}dr, \ \implies \ \frac{4\pi r^{2}dr}{dt} \ = \ \frac{k}{r} \ \implies \ \int4\pi r^{3}dr \ = \ \int k dt\)

\(\displaystyle Hence, \ \pi r^{4} \ = \ kt \ +C, \ \pi \ = \ C, \ so \ \pi r^{4} \ = \ kt \ + \ \pi\)

\(\displaystyle r^{4} \ = \ \frac{kt}{\pi}+1, \ r(t) \ = \ \bigg(\frac{kt}{\pi}+1\bigg)^{1/4}\)

\(\displaystyle r(5) \ = \ 12 \ = \ \bigg(\frac{5k}{\pi}+1\bigg)^{1/4} \ \implies \ k \ = \ 4147\pi\)

\(\displaystyle Therefore, \ r(t) \ = \ (4147t+1)^{1/4}\)