Answer
$50+\frac{c}{2}$
Work Step by Step
Step 1. The profit for selling each backpack is $p=x-c$ and the total profit of selling $n$ backpacks is then $P=np=n(x-c)$
Step 2. Given $n=\frac{a}{x-c}+b(100-x)$, we have $P=a+b(100-x)(x-c)=a+100bx-100bc-bx^2+bcx$
Step 3. Take the derivative to get $P'=100b+bc-2bx$. Let $P'=0$ to get $x=\frac{100+c}{2}=50+\frac{c}{2}$
Step 4. Check $P''=-2b\lt0$ and we know the region is concave down with a maximum. Thus the above $x$ value will give a maximum of the profit.
