The angle of elevation of the top P of a tower from the feet of one person standing due So…

Question

The angle of elevation of the top $P$ of a tower from the feet of one person standing due South of the tower is $45^\circ$ and from the feet of another person standing due west of the tower is $30^\circ$. If the height of the tower is 5 meters, then the distance (in meters) between the two persons is equal to:

Accepted Answer

Let's denote the person standing due south as S and the one standing due west as W. Also, let the tower be at point T.

From person S's perspective, we have a right triangle $	riangle SPT$. The height of the tower PT is given as 5m, which is the opposite side for angle S. The angle at S is $45^\circ$. From the tangent trigonometric ratio, we have :

$
	an{45^\circ} = \frac{PT}{ST} = \frac{5}{ST}
$

Since $	an{45^\circ}=1$, we get $ST = 5$m.

Similarly, from person W's perspective, we have a right triangle $	riangle WPT$. The angle at W is $30^\circ$. From the tangent trigonometric ratio, we have:

$	an{30^\circ} = \frac{PT}{WT} = \frac{5}{WT}$

Since $	an{30^\circ}=\frac{1}{\sqrt{3}}$, we get $WT = 5\sqrt{3}$m.

Since S and W are perpendicular to each other (one is due south and the other is due west), $	riangle SWT$ forms a right triangle.

We can find $SW$ (the distance between the two people) using the Pythagorean theorem:

$
SW^2 = ST^2 + WT^2 = 5^2 + (5\sqrt{3})^2 = 25 + 75 = 100
$

So, $SW = \sqrt{100} = 10$m.

Hence, the correct answer is 10 meters, which corresponds to Option A.

Solutions · 1