Precalculus Practice Test 4 – Step-by-Step Guidance

Q3. To find the distance across a lake, a surveyor took the measurements in the figure shown. Use these measurements to determine how far it is across the lake.

Background

Topic: Law of Sines (Trigonometry in Triangles)

This question tests your ability to apply the Law of Sines to solve for an unknown side in a triangle, given two angles and one side. This is a common application in surveying and navigation.

Key Terms and Formulas

• Law of Sines: For any triangle with sides $a$, $b$, $c$ and opposite angles $A$, $B$, $C$:

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

• Central Angle: The angle at the center of a circle or, in this context, the angle at a triangle vertex.

• Rounding: The final answer should be rounded to the nearest whole number.

Step-by-Step Guidance

1. Identify the triangle in the diagram and label the known sides and angles. Let $a$ be the unknown distance across the lake, and use the given side and angle measurements from the diagram.

2. Write down the Law of Sines formula for the triangle: $\frac{a}{\sin A} = \frac{b}{\sin B}$, where $a$ is the unknown side, $A$ is the angle opposite $a$, $b$ is the known side, and $B$ is the angle opposite $b$.

3. Plug in the known values for $b$ and $B$ (from the diagram), and the value for $A$ (the angle opposite the unknown side $a$).

4. Rearrange the equation to solve for $a$: $a = b \cdot \frac{\sin A}{\sin B}$.

Try solving on your own before revealing the answer!

Final Answer: 817 yards

Using the Law of Sines and the given measurements, the distance across the lake is approximately 817 yards.