During a visit to New York City, Lil decides to estimate the height of the Empire State Building (see figure below). She measures the angle θ of elevation of the spire atop the building as 17°. After walking x = 9.28 ✕ 102 ft closer to the iconic building, she finds the angle to be 20.9°. Use Lil's data to estimate the height h of the Empire State Building. (Enter your answer to at least the nearest
To estimate the height of the Empire State Building, we can use trigonometric ratios and set up a proportion.
Let's start by labeling the given information:
The first angle of elevation (θ1) is 17°.
The second angle of elevation (θ2) is 20.9°.
The distance Lil walks closer to the building (x) is 9.28 * 10^2 ft.
Now, let's proceed with the steps to find the height:
Step 1: Set up a proportion using the tangent ratio.
The tangent of an angle in a right triangle is equal to the ratio of the opposite side to the adjacent side. In this case, the height of the building will be the opposite side, and the distance Lil walks closer will be the adjacent side.
Using the first angle:
tan(θ1) = height of the Empire State Building / x
Using the second angle:
tan(θ2) = height of the Empire State Building / (x - 9.28 * 10^2)
Step 2: Solve the proportion for the height.
We can rewrite the first equation as:
height of the Empire State Building = tan(θ1) * x
Similarly, we can rewrite the second equation as:
height of the Empire State Building = tan(θ2) * (x - 9.28 * 10^2)
Step 3: Calculate the height using the given data.
Substitute the values into the equations. Recall that you must convert the angles to radians when using trigonometric functions.
For the first angle:
height of the Empire State Building = tan(17°) * 9.28 * 10^2 ft
For the second angle:
height of the Empire State Building = tan(20.9°) * (9.28 * 10^2 ft - 9.28 * 10^2 ft)
Simplify the expressions and calculate the values.
Finally, you'll have an estimate of the height of the Empire State Building based on Lil's data.
1st: Split the two angle from the ground to the spire atop into two right triangles
2nd: take tangent of the two angles and multiply them --> tan(17)time tan(20.9)=-6.71696
3rd: 9.28x10^2=928
4th: multiply 928time-6.71696= -6233.3
5th: htan (17)-htan(20.9)=-6233.3
6th: factor out the h-->h(tan(20.9)-tan(17))=-6233.3
7th:h(-5.4)=-6233.3
8th divid-6233.3 by -5.4=1154ft
9th: the answer is1,157 feet
Divide both sides by Tan17:
d = 1.25*(d-928),
d = 4636.4 Ft.
h = d*Tan17 = 4636.4 * Tan17 =
It's easy if you
(a) draw a diagram
(b) review your trig functions
Then it's clear that if the height is h,
h cot17° - h cot20.9° = 928
Now just find h.
Tan17 = h/d.
h = d*Tan17.
Tan20.9 = h/(d-928).
h = (d-928)*Tan20.9.
d*Tan17 = (d-928)*Tan20.9.
d = ?.
h = d*Tan17.