A man flies a kite at a height of 50m. The wind carries the kite horizontally away from him at a rate of?

Question

8m/sec. How fast is the distance between the man and the kite changing when the kite is 130m away from him?
please show steps. thank you.

Answer 1

Well, that isn't how kite-flying works. If the kite is getting farther away, it will be rising vertically as well as moving horizontally.
I hope that this problem isn't from an aeronautical engineering course.

But given the statement of the problem, here's what you do:

Distance from man to kite = sqrt(x^2 + 50^2)
x is the horizontal distance to the kite, 50 is the vertical distance to the kite, and the above expression (the square root) is the hypotenuse of a right triangle, which is the distance from man to kite.

The rate at which the distance is changing can be found by taking the derivative of distance:

d/dx (x^2 + 50^2)^0.5
= 0.5(x^2 + 50^2)^(-0.5)(2x)
= x/(x^2 + 50^2)^0.5)

There are three more steps:
1. Solve for the value of x when the kite is 130 m away:
sqrt(x^2 + 50^2) = 130
(x^2 + 2500) = 16900
x^2 = 14400
x = 120 (also minus 120, but that's an extraneous root, since it doesn't make sense)
2. Substitute the calculated x value into the expression for the rate at which the distance is changing:
d/dx (distance from man to kite) = x/(x^2 + 50^2)^0.5) =
120/(120^2 + 50^2)^0.5) =
120 / (14400 + 2500)^0.5 =
120 / (16900)^0.5 =
120 / 130 =
12/13
3. Since this number is the rate of change in the distance (in meters) for each 1 meter change in x, we have to multiply it by dx/dt (in m/sec) to get the rate of change (in distance) per second. Fortunately, we have been given that dx/dt = 8m/sec.
So the answer to the problem is (12/13) (8 m/sec.) =
96/13 m/sec, which equals about 7.38 m/sec.

Answer 2

your not giving a variable so the kite would stay at the same rate no matter how far away. 8m/sec