A triangle PQR has PQ = 41.6 m, PR = 70 m, and angle PQR = 105 degrees. Calculate angle QPR.

Step 1: Identify the given information.
We are given a triangle PQR with:

• Side $PQ = 41.6 \text{ m}$
• Side $PR = 70 \text{ m}$
• Angle $\angle PQR = 105^\circ$

Part (i): Calculate angle QPR.

To find $\angle QPR$, we first need to find $\angle PRQ$ using the Sine Rule.
The Sine Rule states that for a triangle with sides $a, b, c$ and opposite angles $A, B, C$:

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

Applying this to $\triangle PQR$:

$$\frac{PQ}{\sin(\angle PRQ)} = \frac{PR}{\sin(\angle PQR)}$$

Substitute the known values:

$$\frac{41.6}{\sin(\angle PRQ)} = \frac{70}{\sin 105^\circ}$$

Rearrange to solve for $\sin(\angle PRQ)$:

$$\sin(\angle PRQ) = \frac{41.6 \times \sin 105^\circ}{70}$$

Calculate the value:

$$\sin 105^\circ \approx 0.9659258$$ $$$$