Question

From the top of the house $32m$ high, the angle of elevation if the angle of elevation of the top of the tower is ${45^ \circ }$ and the angle of depression of the foot of the tower is ${30^ \circ }$, then find the height of the tower
$(A)\dfrac{{32}}{{\sqrt 3 }}(\sqrt 3 + 1)meters \\\ (B)32(\sqrt 3 + 1)meters \\\ (C)32\sqrt 3 meters \\\ (D)\dfrac{{32}}{3}(\sqrt 3 + 1) \\\$

Answer 1

The best way to solve this type of question is to draw a diagram and substitute the required information and then look forward to which trigonometric function can provide us with our required detail. We generally use tan trigonometric functions.

Complete step-by-step answer:
The given information that is given is represented below
Height of house = $32m$
Angle of elevation of the top of the tower = ${45^ \circ }$
Angle of depression of the foot of the tower = ${30^ \circ }$
Let us first plot the given information
Let AB denote the height of the house and EG denote the height of the tower which we need to determine and angle of elevation, angle of depression can be represented as follows
The figure of the given information is given below

In the given figure AB denotes the height of the house and the angle of elevation and angle of depression is given in the above figure, Now let us consider that the distance between the foot of the house and the foot of the tower be given as x, as shown in the below figure

Now let us first determine the relation of the of AF with EF
Taking the tangent in the right-angled triangle EFA
$\tan {45^ \circ } = \dfrac{{Altitude}}{{Base}} = \dfrac{{EF}}{x} \\\ EF = x \\\$
Now, Taking the another right angled- triangle AFG
$\tan {30^ \circ } = \dfrac{{Altitude}}{{base}} = \dfrac{{32}}{x} \\\ x = 32\sqrt 3 \\\$
And from the above relation
$y = x = 32\sqrt 3$
So, the height of the tower be given as
$EG = 32 + 32\sqrt 3 = 32(1 + \sqrt 3 )$
This is the required length of the tower
So, the required answer is (B).

Note: In this type of question first we need to determine the value of the relation between the foot of the house to the foot of the tower. The angle of elevation is an angle that is formed between the horizontal line and the line of sight.