On top of the sears building is a tv tower. From a point 300 ft away the bottom of the tower is at an angle of elevation equal to 42.14 degrees and the top of the tower is at an angle of elevation equal to 45.81 degrees. How tall is the tower.
To find the height of the tower, we can use trigonometry and create a right triangle with the tower as the height. Here's how to solve the problem step by step:
Step 1: Draw a diagram:
Visualize the scenario by drawing a diagram. Include the Sears building as a vertical line, the tower as another vertical line on top of the building, and a horizontal line coming out from the bottom of the tower. Label the distance from the point 300 ft away to the bottom of the tower as the base and the height of the tower as the height.
Step 2: Identify the given values:
The problem states that the angle of elevation from the observation point to the bottom of the tower is 42.14 degrees, and the angle of elevation from the observation point to the top of the tower is 45.81 degrees. The distance from the observation point to the bottom of the tower is given as 300 ft.
Step 3: Use trigonometry:
In the right triangle formed by the base, height, and line of sight to the bottom of the tower, we can use the tangent function to relate the angle and the opposite side (height) to the adjacent side (base):
tan(angle) = opposite / adjacent
Let's consider the angle of 42.14 degrees:
tan(42.14) = height / 300
Rearranging the equation to solve for the height:
height = 300 * tan(42.14)
Step 4: Calculate the height:
Using a scientific calculator or an online calculator, evaluate the expression:
height = 300 * tan(42.14) ≈ 300 * 0.9135 ≈ 274.05 ft
Therefore, the height of the tower is approximately 274.05 ft.