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To determine the width of a river from point $A$ to point $B$, a surveyor walks downriver $40 \mathrm{ft}$ along a line perpendicular to $\overline{A B}$ to a new position at point $C$. The surveyor determines that the measure of $\angle A C B$ is $30^{\circ}$. Find the exact width of the river from point $A$ to point $B$. Rationalize denominators if necessary.
The exact width of the river from $A$ and $B$ is $\square \mathrm{ft}$.
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Answer
Final Answer: The exact width of the river from \(A\) and \(B\) is \(\boxed{20}\) ft.
Step 1 ：The problem describes a right triangle with a 30 degree angle. We know that the side opposite the 30 degree angle (the width of the river) is half the length of the hypotenuse. In this case, the hypotenuse is the 40 ft the surveyor walked.
Step 2 ：Therefore, the width of the river should be half of 40 ft, or 20 ft.
Step 3 ：Final Answer: The exact width of the river from \(A\) and \(B\) is \(\boxed{20}\) ft.
