A man 2m tall, walks at the rate of `1 2/3m//s e c` towards a street light which is `5 1/3` m above the ground. At what rate is tip of his shadow moving? At what rate is the length of the shadow changing when he is `3 1/(13)m` from the base of the light?

Explanation

Let `{BC}={xm}, {CE}={ym}` and `frac{d x}{d t}=-frac{5}{3} m / s`.


From `triangle A B E` and `triangle D C E`, we see that


`triangle A B E-triangle D C E`



`therefore frac{A B}{D C}=frac{B E}{C E} Rightarrow frac{frac{16}{3}}{2}=frac{x+y}{y}`


`Rightarrow frac{16}{6}=frac{x+y}{y} `


`Rightarrow 16 y=6 x+6 y Rightarrow 10 y=6 x `


`Rightarrow y=frac{3}{5} x`


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