One afternoon, a couple walks three-fourth of the way around a circular lake, the radius of which is 1.45km. They start at the south side of the lake and head due east at the beginning of their walk. (a) What is the distance they travel? (b) What are the magnitude and direction (relative to due west) of the couple’s displacement?
distance= 1/4* 2PI*radius
they are going NE, so relative to due West, 135 deg.
magnitude= sqrt(2r^2)=r*1.414
To solve this problem, we can break it down into two parts: finding the distance traveled and finding the displacement of the couple.
(a) To find the distance traveled, we need to determine the length of three-fourths of the circumference of the lake. The formula for the circumference of a circle is C = 2πr, where C is the circumference and r is the radius.
Substituting the given radius (1.45 km) into the formula, we have:
C = 2π(1.45 km)
C ≈ 2(3.14)(1.45 km)
C ≈ 9.12 km (rounded to two decimal places)
To find three-fourths of the circumference, we multiply by 3/4:
Distance traveled = (3/4) * 9.12 km
Distance traveled ≈ 6.84 km (rounded to two decimal places)
Therefore, the couple traveled approximately 6.84 km.
(b) The magnitude of the couple's displacement refers to the straight-line distance from their starting point to their ending point. Since they walked three-fourths of the way around the lake, their ending point is at the north side.
To find the magnitude of the displacement, we can use the formula for the length of an arc of a circle. The formula is L = rθ, where L is the length of the arc, r is the radius, and θ is the central angle subtended by the arc.
In this case, since they walked three-fourths of the way around the lake, the central angle is 270 degrees (3/4 of 360 degrees). To convert degrees to radians, we use the formula Radians = (π/180) * degrees.
So, the central angle in radians is:
θ = (π/180) * 270 degrees
θ ≈ 4.71 radians (rounded to two decimal places)
Now, we can find the length of the arc using the formula:
L = 1.45 km * 4.71 radians
L ≈ 6.80 km (rounded to two decimal places)
The magnitude of the couple's displacement is approximately 6.80 km.
To find the direction of the displacement relative to due west, we can use the concept of angles measured counterclockwise from the west direction. Since the couple started due east and ended at the north side, their displacement angle relative to due west is 90 degrees.
Therefore, the magnitude and direction of the couple's displacement are approximately 6.80 km in the direction 90 degrees counterclockwise from due west.