question about similar triangle word problem

Question

A person is walking directly away from a light on an $18$-foot tall pole. At this instant, the person casts a shadow $14$ feet long. If they walk 10 feet farther from the pole, they will cast a shadow $20$ feet long. How tall is the person?

I've tried setting up $2$ unknown variables for two different triangle figures since you don't know the height of the person and (I think) you don't know the initial distance from the pole for the shadow to appear. Here's how I set it up:

$h$ = height of person $x$ = distance for shadow to form

Similar triangle $1$: $(\frac{18}{14}+x)=(\frac{h}{14}) = 252=14h+hx$

Similar triangle $2$: $(\frac{18}{x}+30)=(\frac{h}{20}) = 360=30h+hx$

I'm pretty sure I set the problem up wrong because I can't seem to get the answer for h.

Answer 1

I do not see anything wrong with your workings.

$(hx+30h) - (hx+14h) = 360 - 252$

That gives you $h = 6\frac{3}{4} ft.$

Answer 2

See the image (not to scale!): We have from the smaller triangle: $$(x+14):18 = 14:h$$ and from the big triangle: $$(x+30):18 = 20:h$$ which resolves to $$\begin{cases}x=18\times 14:h - 14\\x=18\times 20:h-30\end{cases}$$ Left hand sides are the same, so right hand sides are equal, too: $$\frac{18\cdot 14}h-14 = \frac{20\cdot 18}h-30$$ Multiply both sides by $h$ to obtain $$18\cdot 14-14h = 20\cdot 18-30h$$ hence $$(30-14)h = (20-14)\cdot 18$$ $$16 h= 6\cdot 18$$ and the answer is: $$h=6\tfrac 34$$