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. Let's break down the steps:
Step 1: Draw a diagram
Draw a right-angled triangle to represent the situation. Label the bottom of the tower as A, the top of the tower as B, and the point where the observer is located as C. Label the angle of elevation at the bottom of the tower as α, and the angle of elevation at the top of the tower as β. Draw a horizontal line from point A to point D, where D is directly below C.
Step 2: Identify known information
Given:
- The distance between point C and the bottom of the tower (AD) is 300 ft.
- The angle of elevation at the bottom of the tower (α) is 42.14 degrees.
- The angle of elevation at the top of the tower (β) is 45.81 degrees.
Step 3: Identify what needs to be found
We need to find the height of the tower, which is the vertical distance AB.
Step 4: Solve
First, let's find the length of AD. We can use the tangent function:
tan(α) = AD/CD
tan(42.14) = AD/300
AD = 300 * tan(42.14)
Next, let's find the length of BD. Using the same approach with the tangent function:
tan(β) = BD/CD
tan(45.81) = BD/300
BD = 300 * tan(45.81)
Lastly, to find the height of the tower (AB), we subtract AD from BD:
AB = BD - AD
Substitute the values we found for AD and BD into the equation to get the final answer.
Remember to convert the angles to radians if the calculator or formula requires it.
So, by following these steps and using the given information, you can determine the height of the tower.