Related rates calculus problem involving shadow lengths A light on the ground is 30 feet away from a building. A 4 foot tall man is walking from the light to the building at a rate of 3 feet per second. He is casting a shadow on the side of the building. At what rate is his shadow shrinking when he is 5 feet from the building? This is my figure: !enter image description here Is this correct? How do I solve this problem? I tried using similar triangles but couldn't succeed. Could someone explain this with a figure? Modified figure: light is on the ground. !enter image description here

Perhaps you can do the drawing. Let $L$ be the location of the light. (It is on the ground!) Let $F$ be the location of the man's feet, and let $H$ be the top of his head. Let $B$ be the bottom of the building. Draw the horizontal line $LB$. Draw the vertical line $FH$ to represent the short thin man.

Draw the line through $L$ and $H$. Let it meet the wall of the building at $S$. Then $BS$ is the length of the shadow.

It is convenient to call the distance $LF$ by the name $x$. We are told the man is walking at $3$ feet per second, so $\frac{dx}{dt}=3$.

Let $y=BS$. Note that triangles $LBS$ and $LFH$ are _similar_ , so $$\frac{y}{30}=\frac{4}{x}.$$ We may want to rewrite this as $$y=\frac{120}{x}.$$ Differentiate. We get $$\frac{dy}{dt}=-\frac{120}{x^2}\frac{dx}{dt}.$$ Now it is almost over.