related rates problems?

Question

1. A lighthouse is located on a small island 3 km away from the nearest point P on a straight shoreline and its light makes four revolutions per minute, How fast is the beam of light moving along the shoreline when it is 1 km from P?

2. A ferris wheel with a radius of 10 m is rotating at a rate of one revolution every 2 minutes. How fast is a rider rising when his seat is 16 m above ground level.

Answer 1

I am answering your first question and I hope you will use this to figure out the second.
Since your light house is 3 km away from the point P, you can draw a right triangle. On one side it is 3, on the other x (where the 1km is supposed to be, we label it x because it is changing), and your hypotenuse.

Now, x/3 = tan y
x=3 tan(y)
differentiate, dx/dt = 3[sec(y)]^2 dy/dt
Now the dx/dt is what you want.

You know sec(y) because you know that from your triangle, one side is 3, the distance between the light and the point P is 1, from that you can calculate the hypotenuse to be squareroot of 10. Sec y = H/A, therefore it is (sqt10)/3.

Now you have dx/dt = 3 [(sqt10)/3)]^2 * dy/dt
dy/dt is given to you in the problem to be 4 times a minute. Since dy/dt is the rate of the change of an angle, its unit is radians/min, and so you have 2pi, which is the radian of one circle, and that happens for 4 times in one minute, so your dy/dt is 2pi * 4/min.

Plug that in now
dx/dt = 3 [(sqt10)/3)]^2 * 8pi
should get something around 83.8 km/min.

Now please solve the second question by your self so you can learn how to do them.

Answer 2

the rider on the ferris wheel was blinded by the light at a pre-determined height and fell out. He sued and the lighthouse was closed down. The ferris wheel was moved to a location with better lighting.

The End.

Answer 3

1.
3 km 1km
______ = _____

4 rpm ?

= 4 x 1 = 4 / 3 = 1 1/3 rpm ..... [ I hope this is correct ;) ]


2.
10m 16m
-------- = ---------
1rp2m ?

= 1rp2m x 16m / 10m