To find the distance across a river, a surveyor chooses points A and B, which are x = 450 ft apart on one side of the river (see the figure). She then chooses a reference point C on the opposite side of the river and finds that ∠BAC ≈ 82° and ∠ABC ≈ 52°. Approximate the distance from A to C. (Round your answer to the nearest foot.)

To solve this problem, we can apply the Law of Sines because we have two angles and the side between them. The Law of Sines states:

\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]

Here, we are given:

- \( AB = 450 \, \text{ft} \) (the distance between points A and B),
- \( \angle BAC = 82^\circ \) (angle at point A),
- \( \angle ABC = 52^\circ \) (angle at point B).

We need to find the distance \( AC \) from point A to point C.

### Step 1: Find \( \angle ACB \)
First, we can find the remaining angle \( \angle ACB \) using the sum of angles in a triangle:

\[
\angle ACB = 180^\circ - \angle BAC - \angle ABC
\]

\[
\angle ACB = 180^\circ - 82^\circ - 52^\circ = 46^\circ
\]

### Step 2: Apply the Law of Sines
Using the Law of Sines, we can set up the following ratio:

\[
\frac{AC}{\sin \angle ABC} = \frac{AB}{\sin \angle ACB}
\]

Substitute the known values:

\[
\frac{AC}{\sin 52^\circ} = \frac{450 \, \text{ft}}{\sin 46^\circ}
\]

### Step 3: Solve for \( AC \)
Now, we solve for \( AC \):

\[
AC = \frac{450 \times \sin 52^\circ}{\sin 46^\circ}
\]

Using a calculator to find the sine values:

\[
\sin 52^\circ \approx 0.7880 \quad \text{and} \quad \sin 46^\circ \approx 0.7193
\]

Now substitute these values into the equation:

\[
AC = \frac{450 \times 0.7880}{0.7193} \approx \frac{354.6}{0.7193} \approx 493 \, \text{ft}
\]

Thus, the approximate distance from point A to point C is **493 feet**.

Would you like more details, or have any questions?

### Related questions:
1. How can we calculate the third angle in any triangle given the other two angles?
2. What is the Law of Sines and how does it apply to triangles?
3. How can we apply the Law of Cosines when the Law of Sines isn't applicable?
4. How do we find the sine of an angle without a calculator?
5. How can we check the accuracy of results when rounding values?

### Tip:
Always verify that the sum of all angles in a triangle equals 180° after solving for unknown angles to catch potential mistakes.

To find the distance across a river, a surveyor chooses points A and B, which are x = 450 ft apart on one side of the river. She then chooses a reference point C on the opposite side of the river and finds that ∠BAC ≈ 82° and ∠ABC ≈ 52°. Approximate the distance from A to C. (Round your answer to the nearest foot.)

Learn how to approximate the distance across a river using the Law of Sines in trigonometry. This problem involves angles of 82° and 52° with a known side length of 450 ft, ideal for students in Grades 10-12. The step-by-step solution explains the use of the Law of Sines and geometry of triangles.

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