Related rates shadow question A $5$ meter lamp is casting a shadow on a $1.8$ meter man walking away at $1.2$ meters a second, how fast is the shadow increasing? I have no idea how to do this, it feels like there is missing information. I know that this is a problem about triangles but there is some weird trick that has to be used since only two heights are known which are both the same part of a triangle.

After $t$ seconds, the man has traveled $1.2t$ meters from the lamp. Let $\mathrm{shadow}(t)$ denote the length of the man's shadow after $t$ seconds.

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Note that triangles $\triangle ABC$ and $\triangle CDE$ are similar. Therefore the ratios between corresponding sides must be equal: $$\frac{BC}{AB}=\frac{DE}{CD}$$ which tells us $$\frac{1.2t}{3.2}=\frac{\mathrm{shadow}(t)}{1.8}$$ so that $$\mathrm{shadow}(t)=\frac{1.2t}{3.2}\times 1.8=\frac{2.16 t}{3.2}=\frac{27t}{40}.$$ Therefore $$\frac{d\,\mathrm{shadow}(t)}{dt}=\frac{d}{dt}\left(\frac{27t}{40}\right)=\frac{27}{40}\,\text{m/s}$$