Chapter 12.8, Problem 41E

(a)

To determine

To sketch:the diagram of the situation of air traffic safety.

Answer to Problem 41E

  

Explanation of Solution

Given:

The architects design the angle of elevation from a point on the ground to the top of the $40$ foot,

Tower is ${56}^0$ .

The angle of the elevation is ${42}^0$ .

Concept used:

The angle of elevation is an angle that is formed between the horizontal line and the line of sight.

If the line of sight is upward from the horizontal line, then the angle formed is an angle of elevation.

Calculation:

Let us suppose that owner of an office building $\left(AB\right)$ wishes to erect a microwave tower $\left(BT\right)$ on the top of building.

Suppose that the angle of the elevation from the point $C$ on the ground to the top of the $40$ feet tower is ${56}^0$ and the angle of the elevation from the same point to the top of the building is ${42}^0$ .

The sketch of the above situation is:

  

Hence, the sketch is drawn.

(b)

To determine

To find:whether the city allowed the building of this tower and the maximum allowed height of any structure is $100$ feet.

Answer to Problem 41E

It is not possible to build the tower on the building.

Explanation of Solution

Given:

The architects design the angle of elevation from a point on the ground to the top of the $40$ foot,

Tower is ${56}^0$ .

The angle of the elevation is ${42}^0$ .

Concept used:

The angle of elevation is an angle that is formed between the horizontal line and the line of sight.

If the line of sight is upward from the horizontal line, then the angle formed is an angle of elevation.

Angle of elevation formula:

The formula for finding the angle of elevation depends on knowing the information such as the measures of the opposite, hypotenuse, and adjacent side to the right angle. If the distance from the object and height of the object is given, then the formula for the angle of the elevation is given by:

Tangent of the angle of elevationis the ratio of the height of the object to the distance from the object.

  $\tan \theta =\frac{\text{opposite side}}{\text{Adjacent side}}$ .

Calculation:

First calculate the height of the building from the triangle $ABC$ .

  $\begin{array}{l}Tan{42}^0=\frac{AB}{AC}.\\ AC=\frac{AB}{Tan{42}^0}\mathrm{.........}\left(i\right)\end{array}$

From the triangle $ACT$ :

  $\begin{array}{l}Tan{56}^0=\frac{40+AB}{AC}.\\ AC=\frac{40+AB}{Tan{56}^0}\mathrm{.........}\left(ii\right)\end{array}$

From equation $\left(i\right)\text{ and }\left(ii\right)$ :

  $\begin{array}{l}\frac{AB}{Tan{42}^0}=\frac{40+AB}{AC}.\\ 1.11AB=0.67\left(40+AB\right).\\ 0.435AB=\mathrm{26.8.}\\ AB=\frac{26.8}{0.435}.\\ AB=\mathrm{61.6.}\end{array}$

Since the height of the building is $62$ feet the total height of the building and the tower is $102$ feet, which is more than maximum allowed height.

Hence, it is not possible to build the tower on the building.