The problem in the image can be broken down into three parts:

• A couple walks three-fourths of the way around a circular lake, with a radius of 2.51 km.
• They start at the west side and head due south.

Let's analyze and solve each part of the question:

Given:

• Radius of the lake $r = 2.51 \, \text{km}$.
• The couple walks three-fourths around the circular path.

Part (a) What is the distance they travel?

The total distance around a circular lake is the circumference of the circle, given by: $C = 2 \pi r$ Since they walk three-fourths of the way, the distance they travel is: $\text{Distance} = \frac{3}{4} \times 2 \pi r$ Substituting the radius $r = 2.51 \, \text{km}$: $\text{Distance} = \frac{3}{4} \times 2 \pi \times 2.51$ $\text{Distance} \approx \frac{3}{4} \times 15.769 \approx 11.827 \, \text{km}$ Thus, the distance they travel is approximately 11.83 km.

Part (b) What is the magnitude of the couple’s displacement?

Displacement is the straight-line distance between the starting point and the ending point. Since they walk three-fourths of the way around the lake, the displacement forms a chord of the circle.

• The central angle corresponding to three-fourths of the way around the circle is: $\theta = \frac{3}{4} \times 360^\circ = 270^\circ$
• The displacement forms a straight line connecting the points that subtend a $270^\circ$ angle. The length of this chord can be calculated using the formula: $\text{Displacement} = 2r \sin\left(\frac{\theta}{2}\right)$ Substituting $r = 2.51 \, \text{km}$ and $\theta = 270^\circ$: $\text{Displacement} = 2 \times 2.51 \times \sin\left(\frac{270^\circ}{2}\right) = 2 \times 2.51 \times \sin(135^\circ)$ Since $\sin(135^\circ) = \sin(45^\circ) = \frac{\sqrt{2}}{2}$, we get: $\text{Displacement} = 5.02 \times \frac{\sqrt{2}}{2} \approx 5.02 \times 0.7071 \approx 3.55 \, \text{km}$ Thus, the magnitude of the displacement is approximately 3.55 km.

Part (c) What is the direction (relative to due east) of the couple’s displacement?

Since they start at the west side and walk three-fourths of the way (270°), they end up on the north side of the lake. Therefore, their displacement points in a southeast direction. To find the exact angle:

• The displacement vector forms an angle of $45^\circ$ south of east. Thus, the direction of the displacement is 45° south of east.

Would you like further clarifications on any step? Here are 5 related questions you might find interesting:

1. How do you calculate the displacement for any fraction of a circular path?
2. What is the relationship between distance traveled and displacement for circular motion?
3. How would the displacement change if they walked half the distance around the lake?
4. What would be the direction of displacement if they walked one-fourth of the way around the lake?
5. How does angular displacement differ from linear displacement?

Tip: In problems involving circular motion, remember that the total distance depends on the arc length, but displacement is always the straight-line distance between two points on the circle.