related rates: changing angle as runner heads for home

[math]

A baseball diamond is a square with sides 90 ft long. A player is going from second to third base at 24 ft/s and an umpire is standing at home plate. Theta is angle formed by line from third base to home plate and line of sight from the umpire to the runner. How fast is theta changing when the runner is 30 ft from third base?

i don't know where to start. divide diamond into right triangles?

 

[galactus]

You have a right triangle. Like a lot of these problems, you have ol' Pythagoras.

The sides of the ball diamond are 90 feet. Let the distance from 2nd base to the runner be x, then the distance from the runner to 3rd is 90-x.

You can use Pythagoras and the triangle highlighted in the diagram to find theta.

The angle can be represented by \(\displaystyle \L\\tan({\theta})=\frac{90-x}{90}\)

Differentiate with respect to time:

\(\displaystyle \L\\sec^{2}({\theta})\frac{d{\theta}}{dt}=\frac{-1}{90}\frac{dx}{dt}\)

When the runner is 30' from 3rd, then \(\displaystyle {\theta}=tan^{-1}(\frac{1}{3})\)

So, we have:

\(\displaystyle \L\\sec^{2}(tan^{-1}(\frac{1}{3}))\frac{d{\theta}}{dt}=24(\frac{-1}{90})\)

Solving for \(\displaystyle \frac{d{\theta}}{dt}\):

\(\displaystyle \H\\\frac{d{\theta}}{dt}=\frac{24(\frac{-1}{90})}{sec^{2}(tan^{-1}(\frac{1}{3}))}=\frac{-6}{25} \;\ rad/sec \;\\)

 

[math]

thanks galactus!!