Question

A 40-foot television antenna stands on the top of a building. From A point on the ground, the angles of elevation of the top and bottom of the antenna, respectively, have measurements of 56 degrees and 42 degrees. How tall is the building?

Answer 1

Hint: In this we are given that a 40-foot television antenna stands on the top of a building. From A point on the ground, the angles of elevation of the top and bottom of the antenna, respectively, have measurements of 56 degrees and 42 degrees. We have to find the height of the building. We can use the trigonometric identity \[\tan \theta =\dfrac{opposite}{adjacent}\], where theta is the given measurements. Using this we can find the height of the building.

Complete step by step answer:
Here we are given a 40-foot television antenna that stands on the top of a building. From A point on the ground, the angles of elevation of the top and bottom of the antenna, respectively, have measurements of 56 degrees and 42 degrees. We have to find the height of the building.
We can now draw the diagram from the given data.

we can write as
 \[\Rightarrow \tan \left( {{42}^{\circ }} \right)=\dfrac{b}{d}\]
Where b is the opposite side and d is the adjacent sides.
We can now write the above step as,
\[\begin{align}
  & \Rightarrow \dfrac{b}{d}=\tan {{42}^{\circ }}=0.900404 \\
 & \Rightarrow d=\dfrac{b}{0.900404}........(1) \\
\end{align}\]
We can now write the next part,
\[\begin{align}
  & \Rightarrow \dfrac{40+b}{d}=\tan {{56}^{\circ }}=1.482561 \\
 & \Rightarrow 40+b=d\times 1.482561...........(2) \\
\end{align}\]
We can now substitute (1) in (2), we get
\[\Rightarrow 40+b=\dfrac{b}{0.900404}\times 1.482561\]
We can now simplify the above step, we get
\[\begin{align}
  & \Rightarrow 40+1b=1.646550766\times b \\
 & \Rightarrow 40=1.646550766\left( b \right)-1\left( b \right) \\
 & \Rightarrow 40=0.64655076\times b \\
 & \Rightarrow b=\dfrac{40}{0.64655076}=61.8667\cong 62 \\
\end{align}\]
Therefore, the height of the building is (approximately) 62 feet.

Note: We should always remember some of the trigonometric formulas such as \[\tan \theta =\dfrac{opposite}{adjacent}\]. We can use the scientific calculators to find some complicated degree values and some fraction steps. We can divide the decimal terms using the calculators.