One afternoon, a couple walks three-fourths of the way around a circular lake, the radius of which is 2.16 km. They start at the west side of the lake and head due south to begin with. (a) What is the distance they travel? (b) What is the magnitude of the couple’s displacement? (c) What is the direction (relative to due east) of the couple’s displacement?

To find the distance the couple travels, we need to calculate the length of three-fourths of the circumference of the circular lake.

The formula for the circumference of a circle is C = 2πr, where "C" is the circumference and "r" is the radius.
In this case, the radius is given as 2.16 km.

So, first, we need to calculate the circumference of the lake.
C = 2π(2.16) ≈ 13.58 km.

Then, we need to find three-fourths of the circumference.
Three-fourths of the circumference = (3/4) * 13.58 km.
= 10.185 km.

Therefore, the couple travels approximately 10.185 km around the lake to complete three-fourths of a revolution.

To find the magnitude of the couple's displacement, we need to find the straight-line distance from the starting point to the ending point. Since they walk three-fourths of the way around the lake, their ending point is on the south side of the lake.

We can use the Pythagorean Theorem to calculate the magnitude of the displacement. The Pythagorean theorem states that the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b) in a right triangle.

In this case, the hypotenuse is the couple's displacement, and the two sides of the triangle will be the horizontal distance (2.16 km) and the vertical distance (10.185 km).

Displacement^2 = (2.16 km)^2 + (10.185 km)^2
= 4.6656 km^2 + 103.692225 km^2
= 108.357 km^2

Taking the square root of 108.357 km^2, we find:
Displacement ≈ √108.357 ≈ 10.411 km.

Therefore, the magnitude of the couple's displacement is approximately 10.411 km.

To find the direction of the couple's displacement relative to due east, we can use trigonometry. Since the displacement is a straight line and the starting point is on the west side of the lake, the angle we need to find is the angle between the positive x-axis (east) and the displacement vector.

If we draw a right triangle, with the horizontal distance (x) as the adjacent side and the vertical distance (y) as the opposite side, we can use the tangent function to find the angle.

tan θ = y / x
tan θ = 10.185 km / 2.16 km
θ ≈ tan^(-1)(10.185 km / 2.16 km)

Using a calculator, we find θ ≈ 77.47 degrees.

Therefore, the direction of the couple's displacement (relative to due east) is approximately 77.47 degrees.

to find the distance multiply radius*2 and then take that answer and multiply it by pi (3.14) to get the total distance around lake then multiply the total by 0.75 to get answer for the distance traveled