### Instruction: For how many real values of $c$ do we have $|3-ci| = 7$?### Response: I notice that $|3-ci|$ is the distance between the complex numbers $3-ci$ and $0$ in the complex plane.
So, I can interpret this equation as asking for the values of $c$ that make $3-ci$ lie on a circle of radius $7$ centered at the origin.
I can also write this equation as $\sqrt{(3-0)^2 + (-c-0)^2} = 7$, by using the formula for the distance between two points.
Squaring both sides, I get $9 + c^2 = 49$.
Subtracting $9$ from both sides, I get $c^2 = 40$.
Taking the square root of both sides, I get $c = \pm \sqrt{40}$.
So, there are two real values of $c$ that satisfy the equation.