Related Rates: Flagpole and Moving Car

[physstudent1]

Homework Statement

A flagpole 40 ft high stands on level ground. A flag is attached to a 120 ft rope passing through a pulley at the top of the flagpole. The other end of the rope is tied to a car at ground level. If the car is driving directly away from the flagpole at 3ft/sec, how fast is the flag rising when the top of the flag is 20 ft off the ground.

Homework Equations

The Attempt at a Solution

I used z for the hypotenuse x for the flat leg and y for the vertical leg, I used the fact that the rope is 120 ft to set up z = 120-y where y is how far the flag has risen. So i got (120-y)^2 = x^2 + y^2 I took the derivative ended up with -240dy/dt = 2x(dx/dt) and plugged knowns into get dy/dt of -2.5 can anyone help?

 

[samee]

First of all wouldn't z=80+y? Because when the flag has not risen at all, then there are 40ft of rope going up the pole, then 80 going down to the car for a total of 120ft. Then when the flag is at it's max height, y=40, then 80+40=120 and all the rope is from the top to the car, right? Maybe that will help you a little . . .

 

[physstudent1]

well i decided to set it up a differnet way z^2 = 40^2 +x^2 since the 40will remain the same shouldn't the changing of z be the same as the changing of the flag going up.

so then 2z(dz/dt)=2x(dx/dt) after plugging in knowns I got 2.75 for dz/dt

can anyone verify this ?

 

[physstudent1]

anyone?

 

[Dick]

Ok. z is hypotenuse, x is horizontal distance. i) What's the height of the flag? ii) What's a relation between z and x in this right triangle?

 

[physstudent1]

the height of the flag is 120-z ? the relation between z and x is where I am kind of confused

 

[Dick]

I would say the height of the flag is 40-(120-z). The rope is 120ft long, the leftover rope after spanning the hypotenuse is 120-z, so draped from the top of a 40ft pole, the height is 40-(120-z). Do you agree? z and x are hypotenuse and leg of a right triangle. The other leg is 40ft. That part should be easy.

 

[physstudent1]

ah yes I do agree with that, I think I am thinking to far into the x and z relationship

 

[physstudent1]

is the way that x is related to z just through the Pythagorean theorem

 

[Dick]

is the way that x is related to z just through the Pythagorean theorem

Absolutely.

 

[physstudent1]

so then is the way I did it before correct

z^2 = x^2 +40^2 then derive to get

2z(dz/dt) = 2x(dx/dt) plug in known information and get 2.75 for dz/dt which would be the same rate of change as the flag going up since its the same rope

 

[Dick]

Yep. That looks ok.

 

[physstudent1]

thanks for the help