# Thread: Confusing Rates of Change

1. ## Confusing Rates of Change

Hi

How do I do this:

A lamp is 40 m above the ground and a stone is dropped from the same height at a point 10 m away. Find the speed of the shadow on the ground given that in t seconds the stone falls $\displaystyle 4.9t^2$ m.

Thanks a lot!

2. Hello Sunyata
Originally Posted by Sunyata
Hi

How do I do this:

A lamp is 40 m above the ground and a stone is dropped from the same height at a point 10 m away. Find the speed of the shadow on the ground given that in t seconds the stone falls $\displaystyle 4.9t^2$ m.

Thanks a lot!
Study the diagram I have attached. The distance $\displaystyle x$ m represents the distance of the shadow from $\displaystyle C$, the point directly below $\displaystyle A$, where the stone was dropped.

Using similar triangles:
$\displaystyle \frac{RC}{BC}=\frac{RQ}{PQ}$
What you need to do now:
Substitute into this equation the values I've given you in the diagram.

Solve for $\displaystyle x$ in terms of $\displaystyle t$.

Differentiate to find $\displaystyle \frac{dx}{dt}$. This represents the speed of the shadow.

Can you complete this now?

3. Ummm am I meant to get 4.9t^2x + 49t^2 = 400???

if so, when i implicitly differentiate that, I can't seem to get the answer...

4. Hello Sunyata
Originally Posted by Sunyata
Ummm am I meant to get 4.9t^2x + 49t^2 = 400???

if so, when i implicitly differentiate that, I can't seem to get the answer...
See step 2 above:
Solve for $\displaystyle x$ in terms of $\displaystyle t$
which gives
$\displaystyle x = \frac{400-49t^2}{4.9t^2}$
$\displaystyle =\frac{400}{4.9t^2}-10$
$\displaystyle \Rightarrow \frac{dx}{dt} = \frac{-800t^{-3}}{4.9}$
So the speed of the shadow has magnitude $\displaystyle \frac{8000}{49t^3}$.