Answer
They should increase the price by $0.75$.
Work Step by Step
$R(x)=(1+0.05x)(4000-80x)=-4x^2+120x+4000$
Let's compare $R(x)=-4x^2+120x+4000$ to $f(x)=ax^2+bx+c$. We can see that a=-4, b=120, c=4000. $a\lt0$, hence the graph opens down, and its vertex is a maximum. The maximum value is at $x=-\frac{b}{2a}=-\frac{120}{2\cdot(-4)}=15.$ Hence the maximum value is $f(15)=-4(15)^2+120(15)+4000=4900.$
Thus they should increase the price by $15\cdot0.05=0.75$.
