One afternoon, a couple walks three-fourths of the way around a circular lake, the radius of which is 1.47 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?
64.2m
To find the answers to these questions, we can use some basic geometry and trigonometry principles. Let's break it down step by step:
(a) To find the distance they travel, we need to calculate the circumference of the circular lake. The formula for the circumference of a circle is given by C = 2πr, where C is the circumference and r is the radius of the circle.
In this case, the radius (r) of the lake is 1.47 km. Plugging this value into the formula, we have C = 2π(1.47) km.
Evaluating the expression, we get C ≈ 2π(1.47) km ≈ 2(3.14)(1.47) km ≈ 9.24 km.
Therefore, the distance the couple travels is approximately 9.24 km.
(b) To find the magnitude of the couple's displacement, we need to determine the straight-line distance between their starting point and ending point. We can achieve this by considering the triangle formed between the center of the circle, the starting point, and the ending point.
The displacement is given by the diameter of the circle. The formula for the diameter of a circle is given by D = 2r, where D is the diameter and r is the radius of the circle.
In this case, the radius (r) of the lake is 1.47 km. Plugging this value into the formula, we have D = 2(1.47) km.
Evaluating the expression, we get D = 2(1.47) km ≈ 2.94 km.
Therefore, the magnitude of the couple's displacement is approximately 2.94 km.
(c) To find the direction of the couple's displacement relative to due east, we need to consider the angle formed between the displacement vector and the due east direction.
Since the couple starts at the west side and head south, the angle between their displacement vector and the due east direction is given by θ = 90 degrees.
Therefore, the direction of the couple's displacement relative to due east is 90 degrees.