1. A person is standing 4.15 km from one end of the river and 5.33 km from the other end of the river as shown below.The angle between both paths is 37^circ . How wide is the river? Draw a diagram to help you. (5 marks)

Here's how to solve this problem using the cosine rule:<br /><br />**1. Diagram:**<br /><br />Draw a triangle. Label the vertices A, B, and C.<br /><br />* Point A represents the person's location.<br />* Point B represents one end of the river.<br />* Point C represents the other end of the river.<br />* The side AB is 4.15 km long.<br />* The side AC is 5.33 km long.<br />* The angle at A (angle BAC) is 37°.<br />* The side BC represents the width of the river (let's call it 'w').<br /><br />**2. Cosine Rule:**<br /><br />The cosine rule states: c² = a² + b² - 2ab * cos(C)<br /><br />In our triangle:<br /><br />* c = w (the width of the river)<br />* a = 5.33 km<br />* b = 4.15 km<br />* C = 37°<br /><br />**3. Calculation:**<br /><br />Substitute the values into the cosine rule:<br /><br />w² = (5.33)² + (4.15)² - 2 * 5.33 * 4.15 * cos(37°)<br />w² = 28.4089 + 17.2225 - 44.229 * cos(37°)<br />w² ≈ 28.4089 + 17.2225 - 35.3066 (cos(37°) ≈ 0.7986)<br />w² ≈ 10.3248<br />w = √10.3248<br />w ≈ 3.21 km<br /><br />**4. Answer:**<br /><br />The width of the river is approximately 3.21 km.<br /><br /><br />**5. Verification:**<br /><br />The answer seems reasonable given the lengths of the other two sides and the angle between them. If the angle were larger, the river would be wider. If the angle were smaller, the river would be narrower. Our result fits within this expectation.<br />