Question

From the top of a church spire 96 m high the angle of depression of two vehicles on the road at the same level as the base of the spire and on the same side of it are $x^{\circ }$ and $y^{\circ }$ , where $\tan x^{\circ }={\displaystyle \frac{1}{4}}$ ​and $\tan y^{\circ }={\displaystyle \frac{1}{7}}$ ​. The distance between the vehicles is:
A. 384 m
B. 672 m
C. 288 m
D. None of these

Answer 1

Hint: To do this question, we will first tell that the angle of depression is equal to the angle of elevation and then we will assume the distance between the first car and the church bottom to be ‘x’ m and the second car and the church bottom to be ‘y’ m. then we will draw a labeled figure according to this information. Then we will write the values of $\tan x^{\circ }$ and $\tan y^{\circ }$ in terms of x and y respectively and keep them equal to their given values. As a result, we will obtain the value of x and y and hence their difference will be equal to the distance between the cars. Thus, we will get our answer.

Complete step by step solution:
Here, we have been given the height of the church spire as 96 m and angles of depression on two points on the same side of the church.
We know that the angle of depression is equal to the angle of elevation. Hence, the angle of elevation of the first vehicle to the top of the church is $x^{\circ }$ and that of the second vehicle is $y^{\circ }$.
Now, let us assume the distance from the bottom of the church spire and the first car to be ‘x’ m and the distance between the bottom of the church spire and the second car to be ‘y’ m.
Now, if we draw and label a figure from the above-mentioned information and assumptions, it will look something as follows:

Thus, the distance between the cars will be equal to the difference in the distances between the church bottom and the first car and the church bottom and the second car which will be equal to $(y-x)m$.
Now, we have been given that $\tan x^{\circ }={\displaystyle \frac{1}{4}}$ and $\tan y^{\circ }={\displaystyle \frac{1}{7}}$ .
We know that in a triangle, the value of tan of an angle is given as:
 $\tan \theta ={\displaystyle \frac{\text{perpendicular}}{\text{base}}}$
Now, if we consider the triangle $\mathrm{\Delta }ABD$ , we can see that:
 $\tan x^{\circ }={\displaystyle \frac{96}{x}}$
But we have been given that $\tan x^{\circ }={\displaystyle \frac{1}{4}}$ . Thus, we can say that:
 ${\displaystyle \frac{96}{x}}={\displaystyle \frac{1}{4}}$
Now, solving this, we get the value of x as:
 $\begin{array}{rl}& {\displaystyle \frac{96}{x}}={\displaystyle \frac{1}{4}}\\ & \Rightarrow 96\times 4=x\\ & \Rightarrow x=384m\end{array}$
Thus, the distance between the first car and the bottom of the church spire is 384m.
Now, if we consider the triangle $\mathrm{\Delta }ABC$ , we can see that:
 $\tan y^{\circ }={\displaystyle \frac{96}{y}}$
But we have been given that $\tan y^{\circ }={\displaystyle \frac{1}{7}}$ . Thus, we can say that:
 ${\displaystyle \frac{96}{y}}={\displaystyle \frac{1}{7}}$
Now, solving this, we get the value of y as:
 $\begin{array}{rl}& {\displaystyle \frac{96}{y}}={\displaystyle \frac{1}{7}}\\ & \Rightarrow 96\times 7=y\\ & \Rightarrow y=672m\end{array}$
Thus, the distance between the second car and the bottom of the church spire is 672m.
Hence, the distance between the cars will be:
 $\begin{array}{rl}& y-x\\ & \Rightarrow 672-384\\ & \therefore 288m\end{array}$
Thus, distance between the cars is 288m.
Hence, option (C) is the correct option.

Note:
 Here, we have taken tan of both the given angles but we could have taken any of the trigonometric ratios as long as the value of it comes out in the terms of x and y. But we took tan of the angles because the given values are also of the tan of the angles and if we would have taken any other ratio, we would have to convert them into the other. Thus, we took tan.