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.
X1=Mike's distance from bottom of tower.
X1+21.2 = Lani's distance from bottom of
tower.
Tan40 = h/X1
h = X1*Tan40
Tan38.5 = h/(X1+21.2)
h = (X1+21.2)*Tan38.5
h = X1*Tan40 = (X1+21.2)*Tan38.5
0.839X1 = 0.795X1 + 16.86
0.044X1 = 16.86
X1 = 383 m.
h = 383 * Tan40 = 322 m.
To solve this problem, we can use trigonometry, specifically the tangent function.
Let's consider the angle of elevation from Mike's position. The tangent of an angle is defined as the ratio of the opposite side to the adjacent side. In this case, the opposite side is the height of the Eiffel Tower, and the adjacent side is the distance between Mike and the tower.
Let's define some variables:
- Let h be the height of the Eiffel Tower.
- Let d be the distance between Mike and the tower.
From Mike's position, the angle of elevation to the top of the Eiffel Tower is 40°. Using the tangent function, we can set up the following equation:
tan(40°) = h / d
Similarly, from Lani's position, the angle of elevation to the top of the Eiffel Tower is 38.5°. Using the same logic, we can set up another equation:
tan(38.5°) = h / (d + 21.2)
Now we have a system of two equations with two variables. We can solve for h by solving this system.
First, divide the first equation by the second equation:
[tan(40°) / tan(38.5°)] = h / d / (d + 21.2)
Now we can substitute the values of the tangent:
[0.8391 / 0.7693] = h / d / (d + 21.2)
Simplifying this expression, we get:
1.0909 ≈ h / d / (d + 21.2)
Now we can cross multiply:
1.0909 * d * (d + 21.2) = h
Simplifying further, we have a quadratic equation:
1.0909 * (d^2 + 21.2d) = h
Solving this equation will give us the height of the Eiffel Tower, h.
However, to round to the nearest meter, we need to have a value for h. For that, we need to know the distance d between Mike and the Eiffel Tower. Unfortunately, the problem does not provide us with that information.
Therefore, without the value of d, we cannot determine the height of the Eiffel Tower.