1. **Problem statement:** Calculate the unknown angle $x$ in triangle $PQR$ where the bearing of $Q$ from $P$ is $132^\circ$, angle $PQR$ is $56^\circ$, and the angle between north and $QR$ at $Q$ is $48^\circ$. 2. **Understanding bearings and angles:** - Bearing is measured clockwise from north. - At point $Q$, the angle between north and $QR$ is $48^\circ$. - The angle $PQR$ is $56^\circ$ between $PQ$ and $QR$. 3. **Find angle $x$:** - The angle $x$ is between $QR$ and the vertical line through $Q$ (north line). - Since the angle between north and $QR$ is $48^\circ$, and the angle between $PQ$ and $QR$ is $56^\circ$, the angle between $PQ$ and north at $Q$ is $48^\circ + 56^\circ = 104^\circ$. - The angle $x$ is the angle between $QR$ and north, so $x = 48^\circ$. **Answer:** $x = 48^\circ$. 1. **Problem statement:** Calculate the distance $RP$ given $PQ = 220$ km, $QR = 360$ km, and angle $PQR = 56^\circ$. 2. **Formula used:** Use the Law of Cosines: $$RP^2 = PQ^2 + QR^2 - 2 \times PQ \times QR \times \cos(\angle PQR)$$ 3. **Substitute values:** $$RP^2 = 220^2 + 360^2 - 2 \times 220 \times 360 \times \cos(56^\circ)$$ 4. **Calculate:** $$RP^2 = 48400 + 129600 - 158400 \times \cos(56^\circ)$$ Calculate $\cos(56^\circ) \approx 0.5592$: $$RP^2 = 178000 - 158400 \times 0.5592 = 178000 - 88530.88 = 89469.12$$ 5. **Find $RP$:** $$RP = \sqrt{89469.12} \approx 299.11$$ **Answer:** $RP \approx 299.11$ km. 1. **Problem statement:** Determine the bearing of $R$ from $P$. 2. **Steps:** - Bearing of $Q$ from $P$ is $132^\circ$. - Angle $PQR = 56^\circ$. - Angle between north and $QR$ at $Q$ is $48^\circ$. 3. **Calculate angle $QPR$ in triangle $PQR$:** Sum of angles in triangle $PQR$ is $180^\circ$: $$\angle P + \angle Q + \angle R = 180^\circ$$ Given $\angle Q = 56^\circ$, $\angle R = x = 48^\circ$ (from part i), so: $$\angle P = 180^\circ - 56^\circ - 48^\circ = 76^\circ$$ 4. **Calculate bearing of $R$ from $P$:** - Bearing of $Q$ from $P$ is $132^\circ$. - Angle at $P$ between $PQ$ and $PR$ is $76^\circ$. - Since bearings are clockwise from north, bearing of $R$ from $P$ is: $$132^\circ - 76^\circ = 56^\circ$$ **Answer:** Bearing of $R$ from $P$ is $56^\circ$.