1. **State the problem:** We have a right triangle $\triangle PQR$ with $\angle R = 90^\circ$, $\angle P = 61^\circ$, and side $p = 98$ inches opposite $\angle P$. We need to find the length of side $r$, the hypotenuse opposite the right angle $R$. 2. **Recall the triangle angle sum rule:** The sum of angles in a triangle is $180^\circ$. Since $\angle R = 90^\circ$ and $\angle P = 61^\circ$, then $$\angle Q = 180^\circ - 90^\circ - 61^\circ = 29^\circ.$$ 3. **Use the sine function:** In a right triangle, the sine of an angle is the ratio of the length of the side opposite the angle to the hypotenuse: $$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}.$$ Here, for $\angle P = 61^\circ$, the opposite side is $p = 98$ inches, and the hypotenuse is $r$. 4. **Set up the equation:** $$\sin(61^\circ) = \frac{98}{r}.$$ 5. **Solve for $r$:** Multiply both sides by $r$: $$r \sin(61^\circ) = 98.$$ Divide both sides by $\sin(61^\circ)$: $$r = \frac{98}{\sin(61^\circ)}.$$ Show cancellation: $$r = \frac{98}{\cancel{\sin(61^\circ)}} \times \frac{\cancel{1}}{\sin(61^\circ)} = \frac{98}{\sin(61^\circ)}.$$ 6. **Calculate the value:** Using $\sin(61^\circ) \approx 0.8746$, $$r \approx \frac{98}{0.8746} \approx 112.0.$$ 7. **Answer:** The length of $r$ to the nearest inch is **112 inches**. Thus, the correct choice is (b) 112 in.