Answer
$$6=c$$
Work Step by Step
Differentiate with respect to $x$
\[
c \sqrt{x}=y
\]
\[
\frac{c}{2 \sqrt{x}}=y^{\prime}
\]
Let's assume that the tangent occurs at $a=x$
since the slope of the line is $\frac{3}{2}, y$ must also be $\frac{3}{2}$
\[
\frac{c}{2 \sqrt{a}}=\frac{3}{2}
\]
Multiply both sides by $2 \sqrt{a}$
\[
c=3 \sqrt{a}
\]
The line will meet the curve if and only if the following equation is true
\[
6+\frac{3}{2} a=c \sqrt{a}
\]
Substitute $c$ = $3 \sqrt{a}$
\[
\begin{array}{c}
6+frac{3}{2} a=3 \sqrt{a} \cdot \sqrt{a} \\
6+ \frac{3}{2} a=3 a
\end{array}
\]
Subtract $\frac{3}{2} a$ from both sides
\[
6=\frac{3}{2} a
\]
\[
4=a
\]
Substitute this in $3 \sqrt{a}=c,$ to get
\[
c=3 \sqrt{4}
\]
$c=6$
