This is a trigonometry problem involving a right triangle. We are tasked with finding the side $PQ$, using the given angles and the side $QR = 22 \, \text{mi}$. Here’s how we can solve it step by step:

Given:

1. $\angle QRP = 48^\circ$
2. $\angle QPR = 26^\circ$
3. Hypotenuse $QR = 22 \, \text{mi}$

Steps to solve:

The unknown side $PQ$ is opposite the $48^\circ$ angle, and the hypotenuse is $QR = 22$.

Using the sine ratio: $\sin(\text{angle}) = \frac{\text{opposite}}{\text{hypotenuse}}$

$\sin(48^\circ) = \frac{PQ}{QR}$

$PQ = QR \cdot \sin(48^\circ)$

Calculation:

1. Find $\sin(48^\circ)$: $\sin(48^\circ) \approx 0.7431$

2. Substitute into the equation: $PQ = 22 \cdot 0.7431 \approx 16.35$

3. Round to the nearest tenth: $PQ \approx 16.4 \, \text{mi}$

Thus, the correct answer is B) 16 mi.

Do you have any questions or need further explanation?

Related Questions:

1. How would you calculate the other side, $PR$, using trigonometric ratios?
2. Can the cosine rule be used to solve this problem? Why or why not?
3. What is the full process to verify this solution using the Pythagorean theorem?
4. How does the sum of angles in a triangle help confirm the correctness of the angles?
5. What is the significance of rounding in real-life applications of trigonometry?

Tip:

Always double-check the angle and side relationship in trigonometric problems to avoid confusion between sine, cosine, and tangent!