Hi!!!
Can help me with 2 little exercises of Related Rates of Change :S?
1)A person walks in a straight line at a speed of 4 ft / s. On the floor, 20 feet away from the road there is a lighthouse that remains directed toward the walker. How fast tour the lighthouse when the subject is 15 feet from the point from where he left?
2)An air traffic controller detects two aircraft at the same altitude with paths which converge to a point, flying at right angles to each other. One of the planes is 150 miles from the point and moves at 450 miles / hour. The other plane is 200 miles from the point and moves at 600 miles per hour.
a) How fast decreases the distance between the two planes?
b) How long has the air traffic controller to make one of the planes take another flight path?.
Thank you!!!!
Can help me with 2 little exercises of Related Rates of Change :S?
1)A person walks in a straight line at a speed of 4 ft / s. On the floor, 20 feet away from the road there is a lighthouse that remains directed toward the walker. How fast tour the lighthouse when the subject is 15 feet from the point from where he left?
2)An air traffic controller detects two aircraft at the same altitude with paths which converge to a point, flying at right angles to each other. One of the planes is 150 miles from the point and moves at 450 miles / hour. The other plane is 200 miles from the point and moves at 600 miles per hour.
a) How fast decreases the distance between the two planes?
b) How long has the air traffic controller to make one of the planes take another flight path?.
Thank you!!!!
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Label the point where their paths are set to converge P.
I will call the distance from P to the first airplane x, and the distance from P to the second airplane y.
dx/dt = -450 mi/hr
dy/dt = -600 mi/hr
If the distance between the two planes is z, then the Pythagorean relationship between them is x² + y² = z², and then we are looking for dz/dt when x = 150 mi and y = 200 mi. Note that when x = 150 and y = 200, z = 250 by Pythagorean theorem.
Take the derivative implicitly with respect to t.
2x dx/dt + 2y dy/dt = 2z dz/dt
Divide by 2:
x dx/dt + y dy/dt = z dz/dt
150(-450) + 200(-600) = 250 (dz/dt)
dz/dt = -750 mph
Since there is only 250 miles between the two airplanes at this instant, and the distance between the planes is decreasing at a rate of 750 mph, then the air traffic controller only has 1/3 hour = 20 minutes to change the path of one of the planes so they don't have a midair collision!
I will call the distance from P to the first airplane x, and the distance from P to the second airplane y.
dx/dt = -450 mi/hr
dy/dt = -600 mi/hr
If the distance between the two planes is z, then the Pythagorean relationship between them is x² + y² = z², and then we are looking for dz/dt when x = 150 mi and y = 200 mi. Note that when x = 150 and y = 200, z = 250 by Pythagorean theorem.
Take the derivative implicitly with respect to t.
2x dx/dt + 2y dy/dt = 2z dz/dt
Divide by 2:
x dx/dt + y dy/dt = z dz/dt
150(-450) + 200(-600) = 250 (dz/dt)
dz/dt = -750 mph
Since there is only 250 miles between the two airplanes at this instant, and the distance between the planes is decreasing at a rate of 750 mph, then the air traffic controller only has 1/3 hour = 20 minutes to change the path of one of the planes so they don't have a midair collision!
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You're welcome!
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You should draw the triangles.