Questions: From a point on level ground 5280 feet (one mile) from the base of a television transmitting tower, the angle of elevation to the top of the tower is 21.1°. Approximate the height of the tower.

Solution Steps

To find the height of the tower, we can use trigonometry. Specifically, we will use the tangent function, which relates the angle of elevation to the opposite side (height of the tower) and the adjacent side (distance from the point to the base of the tower). The formula is:

\[ \text{tan}(\theta) = \frac{\text{opposite}}{\text{adjacent}} \]

Rearranging this formula to solve for the height (opposite side), we get:

\[ \text{height} = \text{tan}(\theta) \times \text{adjacent} \]

We are given the distance from the point on the ground to the base of the tower as 5280 feet and the angle of elevation to the top of the tower as \(21.1^\circ\).

To use trigonometric functions, we need to convert the angle from degrees to radians. The conversion is done using the formula: \[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \] Substituting the given angle: \[ \text{radians} = 21.1 \times \frac{\pi}{180} \approx 0.3683 \]

The tangent of the angle of elevation is the ratio of the height of the tower (opposite side) to the distance from the point to the base of the tower (adjacent side): \[ \tan(\theta) = \frac{\text{height}}{\text{distance}} \] Rearranging to solve for the height: \[ \text{height} = \tan(\theta) \times \text{distance} \]

Substitute the known values into the equation: \[ \text{height} = \tan(0.3683) \times 5280 \approx 2037 \]

Final Answer