Change in radius over time for a spherical ball formula

[aspodkfpo]

Homework Statement

https://www.asi.edu.au/wp-content/uploads/2015/08/NQE_2008_Physics.pdf
Pg 7
Q 12 E)

Relevant Equations

v^r = c dT/dt
P = rG
PV = NkT

Algebra in this answer does not seem to flow right. Firstly, the 16, secondly the n term.
Can someone explain or show me the right answer?

 

[etotheipi]

Since big $V$ is used for voltage, denote the volume with little $v$. You know $v = \frac{4}{3}\pi r^3$, so $$\frac{dv}{dt} = 4\pi r^2 \frac{dr}{dt}$$Now you will require the ideal gas law,$$pv = (rG)v = NkT$$ Use the product rule when differentiating both sides w.r.t. $t$,$$Gr\frac{dv}{dt} + Gv\frac{dr}{dt} = Nk \frac{dT}{dt}$$What do you get if you replace $v$ with $\frac{4}{3}\pi r^3$, and also replace $\frac{dv}{dt}$ with $4\pi r^2 \frac{dr}{dt}$? Also, it is helpful to note that $N = nN_A$.