Mike and Lani stand 21.2 meters apart. From Mike’s position, the angle of elevation to the top of the Eiffel Tower is 40°. From Lani’s position, the angle of elevation to the top of the Eiffel Tower is 38.5°. How many meters high is the Eiffel Tower? Round to the nearest meter.
To find the height of the Eiffel Tower, we can use trigonometry and the concept of similar triangles.
First, let's draw a diagram to visualize the situation:
A (top of Eiffel Tower)
/|
/ |
E (Mike) / |B
/ |
/ |
/ |
/θ1 |θ2
/______|
C (Lani)
In this diagram, A represents the top of the Eiffel Tower, E represents Mike's position, and C represents Lani's position. The angles θ1 and θ2 represent the angles of elevation from Mike and Lani, respectively.
Now, let's consider right triangle ACE. We have the angle θ1, which is 40°, and the side EC, which is the distance between Mike and Lani, given as 21.2 meters. We need to find the height AE, which is the height of the Eiffel Tower.
Using the tangent ratio in right triangle ACE, we can write:
tan(θ1) = AE / EC
Rearranging the equation to solve for AE, we have:
AE = tan(θ1) * EC
Similarly, in right triangle ABC, we have the angle θ2, which is 38.5°, and the side BC, which is also the distance between Mike and Lani (21.2 meters). We need to find the height AB, which is again the height of the Eiffel Tower.
Applying the same procedure, we can write:
AB = tan(θ2) * BC
To find the height of the Eiffel Tower, we need to calculate AE and AB separately using the given values for θ1, θ2, and EC (or BC).
Substituting the values into the equations, we have:
AE = tan(40°) * 21.2
AB = tan(38.5°) * 21.2
Using a calculator, we can evaluate these expressions:
AE ≈ 28.79 (rounded to the nearest hundredth)
AB ≈ 27.89 (rounded to the nearest hundredth)
Finally, to find the height of the Eiffel Tower (AC), we can subtract the height of the shorter triangle from the height of the taller triangle:
AC = AE - AB
AC ≈ 28.79 - 27.89 ≈ 0.9 meters
Therefore, the height of the Eiffel Tower is approximately 0.9 meters (rounded to the nearest meter).