Answer
$40\%$ increase in $V$.
Work Step by Step
Step 1. Given the equation $V=kr^4$, we have $dV=4kr^3dr$
Step 2. With $dr=0.1r$, we have $dV-=4kr^3(0.1r)=0.4kr^4=0.4V$
Step 3. Thus, we obtain $\frac{dV}{V}=0.4=40\%$ which means that a $10\%$ increase in $r$ will result in a $40\%$ increase in $V$.
