Answer
$k=27$
Work Step by Step
Using the definition of $f_{\text {ave}}$:
$f_{a v e}=\frac{1}{-a+b} \int_{a}^{b} f(x) d x$
Thus:
$f_{a v e}=\frac{1}{k} \int_{0}^{k} \sqrt{3 x} d x=\frac{2 \sqrt{k}}{\sqrt{3}}$
$b=k$ and $a=0$ and $\sqrt{3 x}=f(x)$
$6=\frac{2 \sqrt{k}}{\sqrt{3}}$
The average has to equal 6:
$27=\left(\frac{6 \sqrt{3}}{2}\right)^{2}=k$
