Question

Two points, A and B, are 526 apart on a level stretch of road leading to a hill. The angle of elevation of the hilltop from A is 26°30', and the angle of elevation from B is 36°40'. How high is the hill

Answer 1

Answer on Question #81010 - Math - Trigonometry

Question

Two points, A and B, are 526 apart on a level stretch of road leading to a hill. The angle of elevation of the hilltop from A is $26{}^{\circ}30'$, and the angle of elevation from B is $36{}^{\circ}40'$. How high is the hill

Solution

1. Points A and B are the data points in the conditions. Point C is the top of the hill. Point D is the projection of the top of the hill on the line AB

2. $\angle CAD = 26{}^{\circ}30' = 26.5{}^{\circ}; \angle CBD = 36{}^{\circ}40' = 36.6667{}^{\circ}$;

3. If $CD = x$ and $BD = y$ then

$$\tan \angle CAD = \frac{x}{y + 526}; \text{ then } x = \tan \angle CAD * y + \tan \angle CAD * 526$$ $$\tan \angle CBD = \frac{x}{y}; \text{ then } x = \tan \angle CBD * y$$

Since $x = x$ then $\tan \angle CAD * y + \tan \angle CAD * 526 = \tan \angle CBD * y$; $\tan \angle CAD = \tan 26.5{}^{\circ} = 0.50$ and $\tan CBD = \tan 36.6667 = 0.74$ then

$$0.5 * y + 0.5 * 526 = 0.74 * y;$$ $$0.24 * y = 263;$$ $$y = 1095.83;$$

4. Since $\tan \angle CBD = \frac{x}{y}$ then $x = \tan \angle CBD * y$;

$$x = 0.74 * 1095.83 = 821.875$$

**Answer**: The height of the hill is equal to 824.875.

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