Replace $x$ with $3$ in the expression $a(x-1)^3+6$ and set the whole expression equal to $30$.

First compute $3-1$, then cube the result, then move the constant $6$ to the other side before solving for $a$.

In Desmos, type y = a(x-1)^3 + 6. Desmos will prompt you to create a slider for a—add the slider so you can adjust the value of $a$.

On a new line, type (3,30) so that Desmos plots this specific point on the coordinate plane.

Move the slider for $a$ until the graph of $y = a(x-1)^3 + 6$ passes exactly through the point (3,30). The value of $a$ shown on the slider at that moment is the solution.

Because the point $(3,30)$ lies on the graph of $y=f(x)$, its coordinates must satisfy the function. That means when $x=3$, the output is $30$, or in equation form: $f(3)=30$.

The function is defined as $f(x)=a(x-1)^3+6$. Substitute $x=3$ and $f(3)=30$ into this formula:

Now simplify inside the parentheses: $3-1=2$, so

Compute the cube: $2^3=8$. This gives

Subtract $6$ from both sides to isolate the term with $a$:

Finally, divide both sides by $8$:

So the value of $a$ is $3$, which corresponds to choice C.