A street light is at the top of a 15 ft tall pole. A woman 6 ft tall walks away from the pole with a speed of 6 ft/sec along a straight path. How fast is the distance from the base of the pole to the tip of her shadow changing when she is 35 ft from the base of the pole?
I looked up several solutions on yahoo.com, but I don't get how they're coming up with the answer. Something about using the laws of similar triangles? I just can't get the right numbers for webassign. Can someone help me?
Best Answer
Let $x=x(t)$ be the distance at time $t$ of the person from the lightpost. Let $s=s(t)$ be the length of the person's shadow. We want to find information about $s'(t)$. We have information about $x'(t)$.
So we need to find a link between $x$ and $s$.
Draw a picture. Let $T$ be the top of the lightpost, and $B$ the bottom of the lightpost. Let $H$ be the top of the person's head, and $F$ the location of her foot. Finally, let $S$ be the end of her shadow. Join $T$ and $B$ by a line, to represent the pole. Join $H$ and $F$. Join $B$ and $S$. And draw the diagonal $TS$, noting it passes through $H$.
Then Triangles $TBS$ and $HFS$ are similar.
Note that $TB=15$, $HF=6$, $BS=x+s$ and $FS=s$.
Thus by similarity we have $$\frac{15}{6}=\frac{x+s}{s}.$$ Now yoy have found the desired relationship. You may want to simplify a bit before differentiating. I assume you can take over from this point.