Solve Applied Problems Using the Law of Sines
-A surveyor standing 50 meters from the base of a building measures the angle to the top of the building and finds it to be $37 ^ { \circ }$. The surveyor then measures the angle to the top of the radio tower on the building and finds that it is $47 ^ { \circ }$. How tall is the radio tower?
A) $15.94$ meters
B) $6.48$ meters
C) $5.83$ meters
D) $8.82$ meters

Question

Quizplus · Accepted Answer

To solve this problem, we use the Law of Sines to find the height of the building and the combined height of the building and the radio tower, then subtract to find the height of the radio tower alone. First, we find the height of the building using the angle of elevation to the top of the building, which is $37^\circ$. We can set up a right triangle with the height of the building as the opposite side and the distance from the surveyor to the building (50 meters) as the adjacent side. Using the tangent function (since $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$), we find the height of the building: $$ \text{Height of building} = 50 \tan(37^\circ) $$ Then, we do the same for the angle of $47^\circ$ to find the combined height of the building and the radio tower: $$ \text{Combined height} = 50 \tan(47^\circ) $$ The height of the radio tower is the difference between the combined height and the height of the building. Calculating these values gives us the height of the radio tower close to $15.94$ meters, making choice A correct.

Solve Applied Problems Using the Law of Sines -A Surveyor 37∘37 ^ { \circ }37∘

Solve Applied Problems Using the Law of Sines
-A Surveyor $37 ^ { \circ }$