Pumpkin Volume 12

With $R=h$ we have the parametrization $\mathbf{r} \; = \; \left(r \sin p \frac{(n+1)\cos t - \cos((n+1)t)}{n} \,,\, r\sin p \frac{(n+1)\sin t - \sin((n+1)t)}{n}\,,\,r\cos p\right)$ and hence we have the elementary volume element ratio $\begin{aligned} X(r,p,t) & =\; \left| \frac{\partial \mathbf{r}}{\partial r} \cdot \left(\frac{\partial \mathbf{r}}{\partial p} \times \frac{\partial \mathbf{r}}{\partial t}\right) \right| \\ & = \; \frac{2(n+1)(n+2)}{n^2}r^2 \sin p \sin^2 \tfrac12 nt \end{aligned}$ so the pumpkin volume is $\begin{aligned} V_p(R,R,n) & = \; \int_0^R \,dr \int_0^\pi\,dp \int_0^{2\pi}\,dt \, X(r,p,t) \\ & = \; \frac{4(n+1)(n+2)}{3n^2} \pi R^3 \end{aligned}$ so that $\frac{V_p(R,R,n)}{V_s(R)} \; = \; \frac{(n+1)(n+2)}{n^2}$ This makes the correct answer $\tfrac{13 \times 14}{144} = \tfrac{91}{72}$ , so the desired answer is $\boxed{163}$ .