1) Suppose you have a street light at a height 10 meters. You drop a rock vertically so that it hits the ground at a distance 5 meters from the street light. The shadow of the rock moves along the ground. Find the speed of the shadow of the rock at the moment that the rock is 3 meters above the ground and moving with the speed of 5 m/s.

2) You say goodbye to your friend at the intersection of two perpendicular roads. At time t = 0 you drive off North at a (constant) speed v and your friend drives West at a (constant) speed w. How fast is the distance between you and your friend increasing at time t?

What should start with these two?


1) Let $x$ be the distance of the shadow from the spot of landing and $h$ be the height of the stone. From similar triangles,

$$\frac xh = \frac{x+5}{10}, \implies x=\frac{5h}{10-h}$$

Take the time derivatives,

$$x' = \frac{50}{(10-h)^2}h'$$

Given that $h=3m$ and $h'=-5m/s$, the speed of the shadow is $x'=-\frac{250}{49}m/s$.

2) Let $d$ be the distance. From the right triangle,


Take the time derivatives to get


  • $\begingroup$ Thanks Quanto!! You help me alot! $\endgroup$ – DZD Nov 14 '19 at 5:48
  • $\begingroup$ @Donald Zou - Glad to be helpful $\endgroup$ – Quanto Nov 14 '19 at 6:09

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