Lisa is running around the circular track x^2+y^2=10000, whose radius is 100 meters, at 4 meters per second. She starts at point (100,0) and runs in the counterclockwise direction. After 30 minutes of running, what are her coordinates?
30*60 = 1800 seconds
* 4 m/s = 4*1800 meters
2 pi r = 200 pi meters circumference
4 *1800/200 pi = 36/pi= 11.459 circles
= 11 circles + .459 circle
.459 * 360 = 165 degrees
x = 100 cos 165 = -96.6
y = 100 sin 165 = +25.9
To find Lisa's coordinates after 30 minutes of running, we need to determine how far she has traveled on the circular track. Since she is running at a constant speed of 4 meters per second, we can calculate the distance she covers in 30 minutes (or half an hour).
Distance = Speed × Time
Distance = 4 meters/second × 30 minutes
(we need to convert 30 minutes into seconds)
Distance = 4 meters/second × 1800 seconds
Distance = 7200 meters
Now, let's understand the problem geometrically. The equation x^2 + y^2 = 10000 represents a circle with a radius of 100 meters, centered at the origin (0,0). Therefore, she is running on the circumference of this circle.
Since Lisa started at the point (100,0) and is running counterclockwise, we can ascertain that she will cover an arc of 7200 meters in the same counterclockwise direction.
The circumference of a circle is given by the formula:
Circumference = 2πr
Circumference = 2 × π × 100
Circumference = 200π meters
To find the angle subtended by an arc of 7200 meters on the circumference of the circle, we can use the formula:
Angle = (Arc Length / Circumference) × 360°
Angle = (7200 / 200π) × 360°
Angle = 36π°
Since Lisa started at the point (100,0) and she is running counterclockwise, the angle of 36π° will be measured counterclockwise from the positive x-axis.
Now, we can find her new coordinates by rotating the point (100,0) counterclockwise by an angle of 36π°.
Using the formula for rotating a point (x, y) counterclockwise by an angle θ:
x' = x*cos(θ) - y*sin(θ)
y' = x*sin(θ) + y*cos(θ)
Let's substitute the values:
x' = 100*cos(36π) - 0*sin(36π)
x' = 100*cos(36π)
x' ≈ 100*cos(180°)
x' ≈ -100
y' = 100*sin(36π) + 0*cos(36π)
y' = 100*sin(36π)
y' ≈ 100*sin(180°)
y' ≈ 0
Therefore, after 30 minutes of running, Lisa's coordinates would be approximately (-100, 0).