Answer :
To calculate the height of the Sears Tower, we can use trigonometry. When we have an angle of elevation and a distance from the tower, we can use the tangent of that angle to find the height because we are dealing with a right triangle.
The tangent of an angle in a right triangle is the ratio of the opposite side (the height we are looking for) to the adjacent side (the distance from the tower). This can be written as:
tan(θ) = opposite / adjacent
Rearranging the formula to solve for the opposite side (the height of the tower), we get:
opposite = adjacent * tan(θ)
We are given:
θ (angle of elevation) = 53°
adjacent (distance from the tower) = 1093.4 feet
Step-by-step solution:
1. Convert the angle of elevation from degrees to radians if necessary (but in this context, we would normally use the tangent function that takes degrees directly). However, for the sake of being thorough, remember that to convert degrees to radians, you would multiply by π and divide by 180.
2. Use the tangent function to find the ratio of the height to the distance from the tower. So, we find tan(53°).
3. Multiply the distance from the tower by the tangent of the angle to find the height:
Height = 1093.4 feet * tan(53°)
4. Use a calculator to find the tan(53°), which is approximately 1.3270.
5. Multiply the distance by the tan of the angle:
Height ≈ 1093.4 feet * 1.3270
6. Calculating this gives:
Height ≈ 1451.70278 feet
7. Finally, round the height to the nearest foot:
Height ≈ 1452 feet
Therefore, the height of the Sears Tower, when rounded to the nearest foot, is approximately 1452 feet.
