1. **State the problem:** We have two drones flying from points P to Q and Q to R with given bearings and speeds. We need to find: (a) The acute angle $\angle P\hat{Q}R$ formed at point Q by the paths PQ and QR. (b) The distance between the two drones after 4 minutes. 2. **Find the acute angle $\angle P\hat{Q}R$: ** - Bearing of PQ is 115° from North. - Bearing of QR is 215° from North. - The angle between the two paths at Q is the difference between these bearings: $$\theta = |215^\circ - 115^\circ| = 100^\circ$$ - The acute angle is the smaller angle between the two lines, so: $$\angle P\hat{Q}R = 180^\circ - 100^\circ = 80^\circ$$ 3. **Find the distance between drones after 4 minutes:** - Convert 4 minutes to seconds: $$4 \text{ minutes} = 4 \times 60 = 240 \text{ seconds}$$ - Drone A travels from P to Q at 2.5 m/s, starting at P and moving towards Q (distance PQ = 800 m). - Drone B travels from Q to R at 7.5 m/s, starting at Q. - Distance Drone A has traveled after 240 s: $$d_A = 2.5 \times 240 = 600 \text{ m}$$ - Distance Drone B has traveled after 240 s: $$d_B = 7.5 \times 240 = 1800 \text{ m}$$ - Coordinates setup (taking Q as origin): - Vector PQ points at 115°, so vector QP points at 295° (opposite direction). - Position of Drone A relative to Q after 240 s: $$\vec{r}_A = -600 \times (\cos 115^\circ, \sin 115^\circ)$$ (negative because A moves from P to Q, so relative to Q it is towards P) Calculate components: $$\cos 115^\circ = -0.4226, \sin 115^\circ = 0.9063$$ $$\vec{r}_A = -600 \times (-0.4226, 0.9063) = (253.56, -543.78)$$ - Position of Drone B relative to Q after 240 s: Bearing 215° from North: $$\vec{r}_B = 1800 \times (\cos 215^\circ, \sin 215^\circ)$$ Calculate components: $$\cos 215^\circ = -0.8192, \sin 215^\circ = -0.5736$$ $$\vec{r}_B = 1800 \times (-0.8192, -0.5736) = (-1474.56, -1032.48)$$ - Distance between drones after 240 s: $$d = |\vec{r}_A - \vec{r}_B| = \sqrt{(253.56 + 1474.56)^2 + (-543.78 + 1032.48)^2}$$ $$= \sqrt{(1728.12)^2 + (488.7)^2} = \sqrt{2986750 + 238841} = \sqrt{3225591} \approx 1796 \text{ m}$$ 4. **Final answers:** - (a) $\angle P\hat{Q}R = 80^\circ$ - (b) Distance between drones after 4 minutes is approximately 1796 m.