To solve for $x$ in this right triangle, we can use trigonometric ratios.

Step-by-Step Solution

1. Identify the sides relative to the angle:

   • We have a right triangle with a given angle of $31^\circ$.
   • The side opposite this angle (31°) is labeled $x$.
   • The hypotenuse of the triangle is given as $15$.

2. Choose the trigonometric function:

   • Since we know the hypotenuse and are looking for the side opposite the given angle, we can use the sine function, which is defined as: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$
   • Here, $\theta = 31^\circ$, the opposite side is $x$, and the hypotenuse is $15$.

3. Set up the equation: $\sin(31^\circ) = \frac{x}{15}$

4. Solve for $x$:

   • Rearrange the equation to isolate $x$: $x = 15 \cdot \sin(31^\circ)$

5. Calculate $x$:

   • Using a calculator to find $\sin(31^\circ) \approx 0.5150$: $x \approx 15 \cdot 0.5150 = 7.725$

6. Round the answer to the nearest tenth: $x \approx 7.7$

Answer

$x = 7.7$

Would you like more details, or do you have any questions?

Here are some related questions for further practice:

1. How would you solve for the hypotenuse if the opposite side and angle were given?
2. What if the adjacent side and the angle were given—how would you solve for $x$?
3. Can you use cosine or tangent to solve this problem instead? Why or why not?
4. How does the value of $x$ change if the angle increases or decreases?
5. What other trigonometric ratios are useful in solving right triangle problems?

Tip: Always double-check which side (opposite, adjacent, or hypotenuse) you’re working with before choosing a trigonometric function.