So I'm doing my online homework on related rates, which is tedious and confusing to me as it is, and I run into this problem with no idea on how to do it. Can anyone help me understand the steps involved in this problem?
Suppose you have a street light at a height $H$. You drop a rock
vertically so that it hits the ground at a distance $d$ from the
street light. Denote the height of the rock by $h$. The shadow of the
rock moves along the ground. Let $s$ denote the distance of the shadow
from the point where the rock impacts the ground. Of course, $s$ and
$h$ are both functions of time. To enter your answer use the notation
$v$ to denote $h'$:$v=h'$.
Then the speed of the shadow at any time while the rock is in the air
is given by $s'= ???$ (where $s'$ is an expression depending on $h, s,$
$H,$ and $v$ (You will find that $d$ drops out of your calculation.)
Now consider the time at which the rock hits the ground. At that time$h=s=0$.
The speed of the shadow at that time is $s'= ???$ where your answer is an
expression depending on $H, v,$ and $d$.Hint: Use similar triangles and
implicit differentiation. For the second part of the problem you will
need to compute a limit.
Best Answer
As the problem says, you have similar triangles in the diagram. You should be able to write an equation $s=$ some function of $h$. What are the similar triangles? What are the corresponding parts? What is the ratio of the sizes? Now take the derivative of that function with respect to time to get $s'=\frac {ds}{dt}$. You will need the derivative of $h$, which you are told is $v$.