Draw a side view of the cone. You get two similar triangles: one for the whole cone (height $12$, radius $9$) and one for the water (height $x$, radius $r$). How can you relate $r$ and $x$ using a ratio?

Once you have $r$ in terms of $x$, substitute into the cone volume formula $V = \frac{1}{3}\pi r^2 h$ with $h = x$. Be careful not to forget any constant factors.

After substituting and simplifying, compare your final algebraic expression for $V(x)$ to the answer choices and see which one is equivalent.

In Desmos, define the water-surface radius in terms of $x$ using similarity: type r(x) = (3/4)x. Then type V(x) = (1/3)pi*(r(x))^2*x to construct the volume function based on the cone volume formula.

Enter each choice as a separate function, for example A(x) = (3pi/16)x^3, B(x) = pi*(3x/4)^2*x, C(x) = (pi/3)*(3x/4)^2*(12-x), and D(x) = (pi/3)*9^2*x. Use a table (click the gear icon) to evaluate all of them and V(x) at several $x$-values between $0$ and $12$; the correct choice is the one that matches V(x) for all tested values.

Because the cone’s tip is at the bottom and the water fills from the tip upward, the water itself forms a smaller right circular cone inside the original cone.

• The height of this water cone is the depth of the water: $h = x$.
• Let $r$ be the radius of the water surface at height $x$.

So we need to express the volume of this smaller cone in terms of $x$ only.

The side profile of the cone is a triangle. The big cone has:

• Height $12$
• Base radius $9$

The small water cone has:

• Height $x$
• Radius $r$

Because the small and big cones are similar, the ratios of corresponding sides are equal:

Solve for $r$:

So the water surface radius is $r = \frac{3}{4}x$. Now we know both $r$ and $h$ in terms of $x$.

The volume of a cone with radius $r$ and height $h$ is

For the water cone:

• $r = \frac{3}{4}x$
• $h = x$

Substitute these into the volume formula:

This expression matches the structure of one of the answer choices before simplification.

Simplify

First square the radius:

Now substitute back:

Simplify $\frac{9}{48}$:

So

which corresponds to answer choice A.