Let's break down the problem step by step: --- ## 1. **Given Data** - Velocity of A, \( v_A = 80. \) mi/h - Velocity of B, \( v_B = 44. \) mi/h - Angle between roads = \( 70^\circ \) We need: - **Relative velocity of B with respect to A** (\( \vec{v}_{B/A} \)), in **ft/s** and the **direction**. --- ## 2. **Convert Speeds to ft/s** 1 mile = 528 ft; 1 hour = 360 s \[ v_A = 80. \ \text{mi/h} \times \frac{528 \ \text{ft}}{1 \ \text{mi}} \times \frac{1 \ \text{h}}{360 \ \text{s}} = 117.33 \ \text{ft/s} \] \[ v_B = 44. \ \text{mi/h} \times \frac{528 \ \text{ft}}{1 \ \text{mi}} \times \frac{1 \ \text{h}}{360 \ \text{s}} = 64.53 \ \text{ft/s} \] --- ## 3. **Express Velocities as Vectors** Let’s choose the **east** direction (right in the diagram) as the **x-axis** and **north** (up) as the **y-axis**. - \( \vec{v}_A \) (A is moving east): \( \vec{v}_A = (v_A, ) \) - \( \vec{v}_B \) (B is moving south, but road is at 70° north of east): \( \vec{v}_B = v_B \cdot (\cos 70^\circ, -\sin 70^\circ) \) > Note: B's velocity is downwards, so the y-component is negative. --- ### Components: - \( v_A = (117.33, ) \) ft/s - \( v_B = v_B(\cos 70^\circ, -\sin 70^\circ) \) Calculate: \[ \cos 70^\circ = .342, \quad \sin 70^\circ = .9397 \] \[ v_B = (64.53 \times .342, 64.53 \times -.9397) = (22.07, -60.63) \ \text{ft/s} \] --- ## 4. **Relative Velocity Calculation** \[ \vec{v}_{B/A} = \vec{v}_B - \vec{v}_A \] \[ \vec{v}_{B/A} = (22.07 - 117.33, -60.63 - ) = (-95.26, -60.63) \text{ ft/s} \] --- ## 5. **Magnitude of Relative Velocity** \[ |\vec{v}_{B/A}| = \sqrt{(-95.26)^2 + (-60.63)^2} = \sqrt{9074.5 + 3676.} = \sqrt{12750.5} \approx 112.9 \text{ ft/s} \] --- ## 6. **Direction (Angle θ from x-axis)** \[ \tan\theta = \frac{|\text{y-component}|}{|\text{x-component}|} = \frac{60.63}{95.26} \] \[ \theta = \arctan\left(\frac{60.63}{95.26}\right) = \arctan(.6367) \approx 32.6^\circ \] Since both x and y are negative, the vector points **below and to the left** (southwest direction), so the angle is **below the x-axis (east)**. --- ## 7. **Final Answers** - **Magnitude:** \( 112.9 \) ft/s - **Direction:** \( 32.6^\circ \) **below the east axis** --- ## **Summary Table** | Quantity | Value | |-----------------------|--------------| | Relative speed | 112.9 ft/s | | Direction | 32.6° below east | --- ### **Fill in the boxes:** - The relative velocity of B with respect to A is **112.9 ft/s** at an angle of **32.6°** below east. --- **If you need the angle in another reference (e.g., measured from north or south), let me know!**