Trigonometry - worm thread related word problem

Question

The Question

The machine tool diagram shows a symmetric worm thread, in which a circular roller of diameter 1.5 inches sits. Find the amount d that the top of the roller rises above the top of the thread, given the information in the diagram. (Hint: Extend the slanted sides of the thread until they meet at a point.)

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My Understanding

I found w and h using basic trig

For w

$$\sin15°= \frac{\text{opposite}}{\text{hypotenuse}} = \frac{0.75}{w}$$ $$w = \frac{0.75}{\sin15°}$$ $$\sin15°= \frac{\sqrt{6} - \sqrt{2}}{4}$$ $$\Longrightarrow w = 2.897$$

For h $$\tan15° = \frac{0.85}{h}$$ $$h = 3.172$$

$$h-w = 0.2744$$

I've been trying to find angles I could use, but nothing comes to mind, so I'll be grateful if you give me some hints. English is not my first language, so sorry for possible mistakes.

Answer 1

$$d=r-(h-w)=0.75-0.2744=0.4756$$

Answer 2

HINT

From the diagram, we have $h+d=r+w$. Rearrange to find $d$: $$d=...?$$

Can you finish it from there? If you have any questions please don't hesitate to ask.