Question

Jerry bicycles from his dorm to the local fitness center: 3.00 miles east and 2.00 miles north. Cindy's apartment is located 1.50 miles west of Jerry's dorm. If Cindy is able to meet Jerry at the fitness center by bicycling in a straight line, what is the length and direction she must travel?

Short answer

Answer: Cindy must travel 4.92 miles at an angle of 24.44 degrees north of east to meet Jerry directly at the fitness center.

Step-by-step solution

Finding the coordinates of the fitness center and Cindy's initial position

Let's assume Jerry's dorm is at the origin of the coordinate plane (0,0). As Jerry bikes 3.00 miles east (x-axis) and 2.00 miles north (y-axis), the coordinate of the fitness center will be (3.00, 2.00) miles. Cindy's apartment is located 1.50 miles west of Jerry's dorm, so her initial position is at the coordinate (-1.50, 0) miles.

Calculating the distance between the fitness center and Cindy's apartment

Now we need to calculate the distance between the fitness center (3.00, 2.00) and Cindy's apartment (-1.50, 0). The distance formula is given by: distance = sqrt((x2 - x1)^2 + (y2 - y1)^2) Plugging the coordinates into the formula: distance = sqrt((3.00 - (-1.50))^2 + (2.00 - 0)^2) = sqrt(4.5^2 + 2^2) = sqrt(20.25 + 4) = sqrt(24.25) = 4.92 miles Cindy has to travel 4.92 miles in a straight line to meet Jerry at the fitness center.

Calculating the direction Cindy must travel

Now we need to find the direction Cindy must travel to meet Jerry directly at the fitness center. To do this, we can use trigonometry to find the angle between the straight line that connects Cindy's apartment and the fitness center and the positive x-axis (east direction). First, calculate the change in x and y coordinates between the two points: Δx = 3.00 - (-1.50) = 4.5 miles Δy = 2.00 - 0 = 2 miles Now, find the angle by taking the inverse tangent (arctan) of the ratio Δy/Δx: angle = arctan(Δy/Δx) = arctan(2/4.5) = 24.44 degrees north of east So, Cindy must travel 4.92 miles at an angle of 24.44 degrees north of east to meet Jerry directly at the fitness center.