Jerry bicycles from his dorm to the local fitness center: 3.06 miles east and 2.07 miles north. Cindy's apartment is located 1.5 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?
she must go north 2.07
she must go east 1.5+3.06 = 4.56
Tan heading north of east = 2.07/4.56
so heading = 24.4 degrees north of east
which is 90-24.4 = 65.6 true compass heading
distance = sqrt (2.07^2 + 4.56^2)
Well, it seems like Cindy has some catching up to do with Jerry! Let's see how she can get to the fitness center in the most direct way.
Now, if Cindy has to travel 1.5 miles west from Jerry's dorm, she needs to go in the opposite direction. That means she'll have to travel 1.5 miles east.
After reaching Jerry's dorm, she then needs to travel 3.06 miles east to get to the fitness center.
So far, Cindy has traveled a total of 1.5 miles + 3.06 miles = 4.56 miles east from her apartment.
Now, to reach the fitness center, she needs to travel 2.07 miles north.
Combining both distances, Cindy must travel a total of 4.56 miles east and 2.07 miles north to reach the fitness center.
If you're more of a math person, we can also calculate the direct distance using the pythagorean theorem:
√((3.06)^2 + (2.07)^2) = √(9.3636 + 4.2849) = √13.6485 ≈ 3.7 miles.
So, Cindy needs to travel approximately 3.7 miles in a direction slightly northeast to meet Jerry at the fitness center.
Hope that gives you a clear picture! Keep pedaling!
To find the length and direction Cindy must travel to meet Jerry at the fitness center, we can create a right triangle with Jerry's dorm as the starting point, Cindy's apartment as the endpoint, and the fitness center being the third vertex.
First, let's find the distance between Cindy's apartment and the fitness center. Since Jerry's dorm is 3.06 miles east and 2.07 miles north of the fitness center, we can subtract 3.06 miles (Jerry's eastward distance) from the distance Cindy must travel:
Distance from Cindy's apartment to fitness center (horizontal): 3.06 miles - 1.5 miles = 1.56 miles east
Now, let's find the vertical distance between Cindy's apartment and the fitness center:
Distance from Cindy's apartment to fitness center (vertical): 2.07 miles
Using the Pythagorean theorem, we can find the length of the diagonal (direct distance) between Cindy's apartment and the fitness center:
(Distance from Cindy's apartment to fitness center)^2 = (Distance from Cindy's apartment to fitness center (horizontal))^2 + (Distance from Cindy's apartment to fitness center (vertical))^2
Direct distance = √((1.56 miles east)^2 + (2.07 miles)^2)
Direct distance = √(2.4336 + 4.2849)
Direct distance ≈ √6.7185
Direct distance ≈ 2.59 miles
Therefore, Cindy must travel approximately 2.59 miles in a straight line to reach the fitness center. The direction she must travel will be determined by the angle formed by the line connecting her apartment and the fitness center relative to the east-west axis. To determine this, we can use trigonometry.
Angle = arctan((Distance from Cindy's apartment to fitness center (vertical))/(Distance from Cindy's apartment to fitness center (horizontal)))
Angle = arctan(2.07 miles/1.56 miles east)
Angle ≈ arctan(1.32692)
Angle ≈ 52.49 degrees
Therefore, Cindy must travel approximately 2.59 miles at an angle of approximately 52.49 degrees east of north to reach the fitness center.
To determine the length and direction Cindy must travel to meet Jerry at the fitness center, we can use the concept of vector addition.
1. Start by visualizing the scenario on a coordinate plane, with Jerry's dorm at the origin (0, 0). The point representing the fitness center would then be located at (3.06, 2.07). Cindy's apartment is located 1.5 miles west of Jerry's dorm, so it would be at (-1.5, 0).
2. We can calculate the distance between Cindy's apartment and the fitness center using the distance formula:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Distance = √((3.06 - (-1.5))^2 + (2.07 - 0)^2)
= √((4.56)^2 + (2.07)^2)
= √(20.9136 + 4.2849)
= √25.1985
≈ 5.02 miles
Therefore, Cindy needs to travel approximately 5.02 miles to reach the fitness center.
3. To determine the direction Cindy must travel, we can use the concept of vectors. The direction can be calculated by finding the angle formed between the line connecting Cindy's apartment and the fitness center and the positive x-axis.
Angle = arctan((y2 - y1) / (x2 - x1))
Angle = arctan((2.07 - 0) / (3.06 - (-1.5)))
= arctan(2.07 / 4.56)
≈ 25.66 degrees
Therefore, Cindy needs to travel in a direction approximately 25.66 degrees east of the positive x-axis to reach the fitness center.