Chapter 5.3, Problem 46E

Problems 41-48 refer to the bounded feasible region with corner points $O=\left(0,0\right),A=\left(0,5\right),B=\left(4,3\right)$, and $C=\left(5,0\right)$ that is determined by the system of inequalities

$\begin{array}{c}x+2y\le 10\\ 3x+y\le 15\\ x,y\ge 0\end{array}$

If $P=ax+10y$, find all numbers $a$ such that the minimum value of $P$ occurs at only $C$.