In triangle ABC, angle A = 45 degrees, angle B = 65 degrees, and side a = 15.05. Us...

1. Identify the problem type: This is a trigonometry problem involving a triangle where we need to find a side length using the Law of Sines.

2. Recall the Law of Sines: The Law of Sines states that for any triangle ABC, the ratio of the length of a side to the sine of its opposite angle is constant:
$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$

3. Identify the given information: We are given:
- Angle $A = 45^{\circ}$
- Angle $B = 65^{\circ}$
- Side $a = 15.05$
We need to find side $b$.

4. Set up the equation using the Law of Sines: We will use the part of the Law of Sines that relates sides $a$ and $b$ to their opposite angles $A$ and $B$:
$$\frac{a}{\sin A} = \frac{b}{\sin B}$$

5. Substitute the given values into the equation:
$$\frac{15.05}{\sin 45^{\circ}} = \frac{b}{\sin 65^{\circ}}$$

6. Solve for $b$: To isolate $b$, multiply both sides of the equation by $\sin 65^{\circ}$:
$$b = \frac{15.05 \times \sin 65^{\circ}}{\sin 45^{\circ}}$$

7. Calculate the values of the sines:
- $\sin 65^{\circ} \approx 0.9063$
- $\sin 45^{\circ} \approx 0.7071$

8. Perform the calculation:
$$b = \frac{15.05 \times 0.9063}{0.7071}$$
$$b = \frac{13.640315}{0.7071}$$
$$b \approx 19.2902$$

9. Round the answer to the nearest tenth: The calculated value of $b$ is approximately $19.2902$. Rounding to the nearest tenth, we look at the hundredths digit, which is 9. Since it's 5 or greater, we round up the tenths digit.
$$b \approx 19.3$$