Problem

In $I_n=\int_0^k x^n(k-x)^{1/2}\,dx \qquad n\ge 0$

where $k$ is a positive constant.

(a) Show that

$$I_n=\frac{2kn}{3+2n}I_{n-1}\qquad n\ge 1$$

Given that

$$\int_0^k x^2(k-x)^{1/2}\,dx=\frac{9\sqrt3}{280}$$

(b) use the result in part (a) to determine the exact value of $k$.