Hi guys !

There's a related rate problem I really can't figure out. I think I have an idea, but I'm not too sure about it. Heres the situation :

A ball is dropped at t=0 at 15 m high, 9 m from a lamp. This lamp has a height of 15 m too. Knowing that the height of the ball can be known with the formula y(t)=15-4.9t², at what speed goes the shadow of the ball at t=0.5 (ie. 0.5s after being dropped)

Here is my drawing of the situation.

I called x the distance between under the ball and the shadow of it. I first calculated what height the ball would be after 0.5s, which gave me :

y(0.5) = 15-4.9(0.5)² = 13.775 m

Then, I decided to calculate its speed after at this moment. This would logically be the rate dy/dt, I thought. It gave me :

dy/dt = (13.775-15)/(0.5) = -2.45 m/s

I tried to figure out how I could relate y (height of the ball) and x, and the only thing I thought of was : Let's use similar triangle rules ! The biggest triangle was 15 m*(9+x) m and the smallest, y*x. I used the following formula :

x/(x+9) = y/15

I isolated x :

x = (9y)/(15-y)

And then, I derivated it in function of the time.

dx/dt = [135(dy/dt)]/[(15-y)²]

I knew that dy/dt = -2.45 m/s and that y = 13.775. So the final answer was :

dx/dt = -220.4 m/s

So, at t=0.5, the shadow of the ball moves toward the lamp at the speed of 220.4 m/s.

Did I do anything wrong ? Is the answer correct or logical/plausible ?

Thanks a lot !

(Sorry, I don't know anything about LaTEX)