Find the hypotenuse of a right triangle given one leg and two angles.

1. Identify the problem type: This is a trigonometry problem involving a right triangle. We are given one leg and two angles, and we need to find the length of the hypotenuse.

1. Analyze the given information: We have a right triangle. One angle is $90^\circ$. Another angle is given as $60^\circ$, and the third angle is $30^\circ$ (since the sum of angles in a triangle is $180^\circ$, $180^\circ - 90^\circ - 60^\circ = 30^\circ$). The side opposite the $60^\circ$ angle has a length of $4\sqrt{3}$ ft. The side labeled $f$ is the hypotenuse (opposite the $90^\circ$ angle).

1. Choose a trigonometric ratio: We can use the sine, cosine, or tangent function. Since we know the side opposite the $60^\circ$ angle and we want to find the hypotenuse, the sine function is appropriate. The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$.

4. Set up the equation: Using the $60^\circ$ angle, the opposite side is $4\sqrt{3}$ ft, and the hypotenuse is $f$. So, we have:
$$\sin(60^\circ) = \frac{4\sqrt{3}}{f}$$

5. Solve for f: We know that $\sin(60^\circ) = \frac{\sqrt{3}}{2}$. Substitute this value into the equation:
$$\frac{\sqrt{3}}{2} = \frac{4\sqrt{3}}{f}$$
To solve for $f$, we can cross-multiply:
$$f \cdot \sqrt{3} = 2 \cdot 4\sqrt{3}$$
$$f\sqrt{3} = 8\sqrt{3}$$
Now, divide both sides by $\sqrt{3}$:
$$f = \frac{8\sqrt{3}}{\sqrt{3}}$$
$$f = 8$$

Alternatively, we could use the $30^\circ$ angle. The side adjacent to the $30^\circ$ angle is $4\sqrt{3}$ ft, and the hypotenuse is $f$. The cosine of an angle is the ratio of the adjacent side to the hypotenuse: $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$.
$$\cos(30^\circ) = \frac{4\sqrt{3}}{f}$$
We know that $\cos(30^\circ) = \frac{\sqrt{3}}{2}$. Substitute this value:
$$\frac{\sqrt{3}}{2} = \frac{4\sqrt{3}}{f}$$
This leads to the same equation as before, and thus the same result for $f$.

1. State the final answer: The length of the hypotenuse $f$ is 8 feet.