After you set up the equation, divide both sides by $500$ to get the expression with $3^{t/4}$ by itself. What number do you get on the right side?

Rewrite the number on the right side as $3$ to some power, then set the exponents equal and solve for $t$.

In Desmos, type y = 500 * 3^(x/4) to graph the function that models the number of bacteria over time (with $x$ representing hours).

On a new line, type y = 13500 to graph a horizontal line representing $13{,}500$ bacteria.

Use the intersection tool (or tap where the two graphs cross) to find the point where the curve and the horizontal line meet. The $x$-coordinate of this intersection is the time, in hours, when the culture first contains exactly $13{,}500$ bacteria.

We are told the number of bacteria after $t$ hours is

and we want the time when the culture first contains exactly $13{,}500$ bacteria.

So set $P(t)$ equal to $13{,}500$:

Divide both sides of the equation by $500$ to get rid of the coefficient:

Simplify the right-hand side:

so

Notice that $27$ is a power of $3$:

So we have

When two powers with the same base are equal, their exponents must be equal, so set the exponents equal:

Solve $\dfrac{t}{4} = 3$ by multiplying both sides by $4$:

So the culture will first contain exactly $13{,}500$ bacteria at $t=12$ hours, which corresponds to answer choice C.