A vertical tower stands on a horizontal plane. From a point on the ground 18 meters away from the base of the tower, the angle of elevation to the top of the tower is 30 degrees. Find the height of the tower.

Step 1: Identify the given information and unknown

We are given the distance from the base of the tower to the observation point, which is 18 meters. The angle of elevation from this point to the top of the tower is 30 degrees. We need to find the height of the tower, which we can denote as $h$. This scenario forms a right-angled triangle.

Step 2: Choose the appropriate trigonometric ratio

In the right-angled triangle formed, the height of the tower is the side opposite to the angle of elevation, and the distance from the base is the side adjacent to the angle of elevation. The tangent function relates the opposite side to the adjacent side.

Step 3: Substitute the values into the tangent equation

Now we substitute the given angle of elevation, $30^\circ$, and the adjacent side, 18 meters, into the tangent formula. The unknown height $h$ is the opposite side.

Step 4: Solve for the height of the tower

We know that $\tan(30^\circ) = \frac{1}{\sqrt{3}}$. Multiplying both sides by 18, we get $h = 18 \times \frac{1}{\sqrt{3}}$. To rationalize the denominator, we multiply the numerator and denominator by $\sqrt{3}$, which simplifies to $6\sqrt{3}$ meters.

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