Solve the problem.
-A surveyor standing 56 meters from the base of a building measures the angle to the top of the building and finds it to be $39 ^ { \circ }$. The surveyor then measures the angle to the top of the radio tower on the building and finds that it is $49 ^ { \circ }$. How tall is the radio tower?
A) $19.07 \mathrm {~m}$
B) $9.87 \mathrm {~m}$
C) $6.78 \mathrm {~m}$
D) $7.02 \mathrm {~m}$

Question

Quizplus · Accepted Answer

To find the height of the radio tower, we first calculate the total height from the surveyor to the top of the radio tower using the tangent of the 49° angle, and then subtract the height from the surveyor to the top of the building using the tangent of the 39° angle. The tangent of an angle in a right triangle is the ratio of the opposite side (height in this case) to the adjacent side (distance from the surveyor to the building, which is 56 meters). For the building height (H1):$$ \tan(39^\circ) = \frac{H1}{56} $$$$ H1 = 56 \cdot \tan(39^\circ) $$For the total height including the radio tower (H2):$$ \tan(49^\circ) = \frac{H2}{56} $$$$ H2 = 56 \cdot \tan(49^\circ) $$The height of the radio tower is the difference between H2 and H1:$$ \text{Height of the radio tower} = H2 - H1 $$$$ = 56 \cdot \tan(49^\circ) - 56 \cdot \tan(39^\circ) $$Calculating this gives a height of approximately 19.07 meters, which matches choice A.

Solve the Problem 39∘39 ^ { \circ }39∘ the Surveyor Then Measures the Angle to the Top

Solve the Problem $39 ^ { \circ }$ the Surveyor Then Measures the Angle to the Top