Related Rates, find d(theta)/dt

[kingdomof]

Homework Statement

A plane is approaching an observer from an altitude of 5 mi at a dx/dt of 600 mi/h. Find the d(theta)/dt when theta is 30 degrees, 60 degrees, and 75 degrees.

Homework Equations

tan(theta) = x/y
5csc(theta) = r

The Attempt at a Solution

For my d(theta)/dt I had the formula of 3000/(5csc(theta))^2

The answers I had were wrong according to the text.

Please help.

 

[Mark44]

Homework Statement

A plane is approaching an observer from an altitude of 5 mi at a dx/dt of 600 mi/h. Find the d(theta)/dt when theta is 30 degrees, 60 degrees, and 75 degrees.

Homework Equations

tan(theta) = x/y
5csc(theta) = r

The Attempt at a Solution

For my d(theta)/dt I had the formula of 3000/(5csc(theta))^2

The answers I had were wrong according to the text.

Please help.

You have too many variables. All you need are two: one for the horizontal distance and one for the angle.

I presume that y is the altitude of the plane, which is given and is not changing. I'm guessing that r is the length of the hypotenuse of the right triangle. Keep in mind that if x is the horizontal distance, it is decreasing because the plane is approaching the observer, so dx/dt will necessarily be negative.

I have no idea how you got what you have for d(theta)/dt.

 

[HallsofIvy]

All you need is $tan(\theta)= height/distance$. You are told that the height is always 5 mi and that the distance is decreasing at 600 mi/hr.

With $\theta= arctan(500/x)$, what is $d\theta/dx$ and from that what is $d\theta/dt$?