How Fast is the Car Traveling Based on Angle Change?

[avr10]

Homework Statement

You are 150 feet away from a road. Looking down the road, you notice a car heading in your direction such that the angle formed by your line of vision to the car and the road is 45 degrees, and this angle is increasing at a rate of 10 degrees per second. How fast is the car traveling?

Homework Equations

The Attempt at a Solution

So I drew the diagram such that I'm standing on the positive side of the x-axis, and such that the car is heading down the y-axis (the road) toward the origin. I'm given \frac {d\theta}{dt} = 10, and I want to find \frac {dy}{dt}.

What I did was set \theta = \arctan { \frac {150}{150-\frac{dy}{dt}*t}} then tried to take the derivative of that...but the \frac {dy}{dt} term is leaving me confused as to how to derive such a thing. Ideas?

 

[HallsofIvy]

I think it would be simpler to just say tan(\theta)= y/150 (I don't know where you got that "120") and use implicit differentiation:
sec^2(\theta)\frac{d\theta}{dt}= \frac{1}{150}\frac{dy}{dt}

 

[avr10]

Thanks; the 120 was a typo which I now fixed. When I do that (I used \tan{\theta} = 150/y), I end up getting something like 3000 ft/s, which I think there is something wrong with... why am I getting such a big number?

 

[HallsofIvy]

Remember that the derivative of tan(x) is sec²(x) when x is measured in radians. sec(45)= \sqrt{2} and you are given that d\theta/dt= 10 degrees= (\pi/180)*10 radians so you should get dy/dt= (150)(2)(\pi/18) which is about 52.2