# Calculus - Related rates problem.

Question:

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?

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,

$$d(t)=\sqrt{(vt)^2+(wt)^2}$$

Take the time derivatives to get

$$d'=\frac{v^2+w^2}{\sqrt{v^2+w^2}}=\sqrt{v^2+w^2}$$

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