Question

A road 9 miles long runs from a point P to a point O on a straight beach, making an angle of 32° with the beach. Two other straight roads, each 6 miles long, lead from P to the beach. How far is it from O along the beach to the nearer of these?

Answer 1

Answer on Question #77647 – Math – Trigonometry

Question

A road 9 miles long runs from a point P to a point O on a straight beach, making an angle of $32{}^{\circ}$ with the beach. Two other straight roads, each 6 miles long, lead from P to the beach. How far is it from O along the beach to the nearer of these?

Solution

Let X and Y be the ends of the two roads leading to the beach. $\mathrm{PX} = \mathrm{PY} = 6$. The question asks to find OX.

Let Z be the middle of XY. Then $\mathrm{OX} = \mathrm{OZ} - \mathrm{ZX}$.

$$\mathrm{OZ} = \mathrm{OP} \cos \angle \mathrm{POZ} = 9 \cos 32{}^{\circ}$$ $$\mathrm{PZ} = \mathrm{OP} \sin \angle \mathrm{POZ} = 9 \sin 32{}^{\circ}, \quad \mathrm{PX} = 6 \Rightarrow \mathrm{XZ}^2 = \mathrm{PX}^2 - \mathrm{PZ}^2 = 36 - 81 \sin^2 32{}^{\circ}$$

Then $\mathrm{OX} = 9 \cos 32{}^{\circ} - \sqrt{36 - 81 \sin^2 32{}^{\circ}}$.

Using calculator $\mathrm{OX} = 3.992$ miles.

**Answer**: 3.992 miles.

Answer provided by https://www.AssignmentExpert.com