Question

a 475 - foot tall monument is located in the distance. from a window in a building, a person determines that the angle of elevation to the top of the monument is 19°, and the angle of depression is 4° to the bottom of the monument. how far is the person from the monument?
round to the nearest whole number.
feet
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Explanation:

Step1: Set up equations using tangent function

Let the distance from the person to the monument be \(x\) feet.
Let the height from the window to the top of the monument be \(h_1\) and to the bottom be \(h_2\). We know that \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\).
For the angle of elevation \(\theta = 19^{\circ}\), \(h_1=x\tan19^{\circ}\). For the angle of depression \(\theta = 4^{\circ}\), \(h_2=x\tan4^{\circ}\). And \(h_1 + h_2=475\).

Step2: Substitute and solve for \(x\)

Substitute \(h_1\) and \(h_2\) into \(h_1 + h_2 = 475\):
\(x\tan19^{\circ}+x\tan4^{\circ}=475\).
Factor out \(x\): \(x(\tan19^{\circ}+\tan4^{\circ}) = 475\).
We know that \(\tan19^{\circ}\approx0.3443\) and \(\tan4^{\circ}\approx0.0699\).
Then \(x=\frac{475}{\tan19^{\circ}+\tan4^{\circ}}=\frac{475}{0.3443 + 0.0699}\).
\(x=\frac{475}{0.4142}\approx1147\).

Answer:

\(1147\)