# Dodecahedron and Sphere Related Rates Problem - Calculus

I was messing around with creating a tough (in my opinion) related rates problem for fun. I ended up with the following problem:

A sphere is inscribed inside a regular dodecahedron. The sphere’s surface area starts increasing at a rate of $e^{a+t}$, where a is the edge length of the dodecahedron and t is the time that has elapsed. The dodecahedron grows with the sphere so that the sphere is always inscribed inside of it. At the moment the shapes start expanding, the dodecahedron’s edge measures 1 unit. At what rate, in terms of a, is the volume of the space between the sphere and dodecahedron changing?

(when t = 0, a = 1)

I thought I solved it, when I realized early on I took the derivative of an expression instead of the integral. Now, I'm stuck on integrating $e^{a+t}$ dt (I put down $e^{a+t}(\frac{da}{dt}+1) + C$). I don't believe it's a double integral because a and t can't be just any values, a depends on t. I thought about getting t in terms of a, but I couldn't seem to do that. I wrote up the problem along with my presumably incorrect solution in a Word document, which I've attached screen shots of. One other thing is confusing me. I believe I messed up, but later on I preform a check that should give me e and it does to 10+ decimals places. Let me know if you think I went awry and the check is a coincidence or somehow I took the integral correctly. Any help would be greatly appreciated, I've been trying to figure this out for hours. Thank you!

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• I don't understand all the work you do after the middle of page 3. Just before the middle you have an equation of the form $\frac{da}{dt}=C\frac{e^ae^t}{a}$ where $C$ is a constant, and you correctly say you need $t$ in terms of $a$. But separating variables gives $ae^{-a}\,da=Ce^t\,dt$, which means $-e^{-a}(a+1)=Ce^t+D$. From here you can use the initial condition to solve for $t$. – symplectomorphic Jan 7 '18 at 22:39
• Thank you, symplectomorphic. This is exactly what I was looking for. I was able to derive an answer using your technique. – B. Freeman Jan 14 '18 at 19:00