Lil decides to estimate the height of the Empire State Building. She measures the angle θ of elevation of the spire atop the building as 19°. After walking x = 9.10 ✕ 102 ft closer to the building, she finds the angle to be 23.8°. Use Lil's data to estimate the height h of the Empire State Building.
1st measure __ h / tan(19º) = d
2nd measure
__ h / tan(28.3º) = d - 910
substitute for d and solve for h
To estimate the height of the Empire State Building, we can use the concept of trigonometry. We can set up a triangle with the height of the building as one side and the distance between Lil's two positions as the base.
Let's call the height of the Empire State Building "h" and the distance from Lil's initial position to the base of the building "d".
From Lil's first position, she measures the angle of elevation to be 19°. This means that in the triangle formed, the angle θ1 opposite to the base distance "d" is 19°.
From Lil's second position, after walking x = 9.10 ✕ 102 ft closer to the building, she measures the angle of elevation to be 23.8°. In this case, the angle θ2 opposite to the base distance "d-x" is 23.8°.
Now, we can set up two equations based on the given information:
Equation 1: tan(θ1) = h / d
Equation 2: tan(θ2) = h / (d - x)
We know the values of θ1 (19°), θ2 (23.8°), and x (9.10 ✕ 102 ft), so we can substitute these values into the equations to solve for the height "h".
Using Equation 1, we have:
tan(19°) = h / d
Using Equation 2, we have:
tan(23.8°) = h / (d - x)
We can rearrange these equations to solve for "h" in terms of "d" and solve the system of equations.
First, let's solve Equation 1 for "d":
d = h / tan(19°)
Next, let's solve Equation 2 for "d":
d - x = h / tan(23.8°)
Substituting the value of "d" from Equation 1 into Equation 2, we get:
h / tan(19°) - x = h / tan(23.8°)
Now, we can solve for "h" by isolating it on one side of the equation:
h / tan(19°) - h / tan(23.8°) = x
To simplify further, we can factor out "h":
h * (1 / tan(19°) - 1 / tan(23.8°)) = x
Finally, we can solve for "h" by dividing both sides of the equation by the expression in parentheses:
h = x / (1 / tan(19°) - 1 / tan(23.8°))
Now, we can substitute the given values of "x" and the angle measurements to find the estimated height "h" of the Empire State Building.
Draw a diagram.
Review the definition of the cotangent.
If the height is h, then
h cot19° - h cot23.8° = 910