A vertical tower stands on a horizontal plane. From a point on the ground 17 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 17 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 $h$.

Step 2: Formulate the trigonometric relationship

The situation forms a right-angled triangle where 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. The tangent function relates the opposite side to the adjacent side.

Step 3: Substitute the value of $\tan(30^\circ)$

We know that the value of $\tan(30^\circ)$ is $\frac{1}{\sqrt{3}}$. We substitute this value into our equation.

Step 4: Solve for the height of the tower

To find $h$, we multiply both sides of the equation by 17. We then rationalize the denominator by multiplying the numerator and denominator by $\sqrt{3}$ to get the final height.

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