Question

Assertion (A)
A person standing on the bank of a river observes that the angle of elevation of the top of a tree standing on the opposite bank is 60°. When he moves 40 m away from the bank, he finds the angle of elevation to be 30°. Then, the height of the tree is $20\sqrt{3}\mathrm{m}.$

Reason (R)
The angle of elevation of the top of a tower as observed from a point on the ground is α and on moving a metres towards the tower, the angle of elevation is β. Then, the height of the tower is
$\frac{\alpha \tan \alpha \tan \beta }{(\tan \beta -\tan \alpha )}.$

(a) Both Assertion (A) and Reason (R) are true and (R) is a correct explanation of Assertion (A).
(b) Both Assertion (A) and Reason (R) are true but Reason (R) is not a correct explanation of Assertion (A).
(c) Assertion (A) is true and Reason (R) is false.
(d) Assertion (A) is false and Reason (R) is true.

Answer 1

Let $AB$ be the tree on the opposite side of the bank of the river and $C$ be the position of the man standing on the bank of the river.
Thus, we have:
$CD\hspace{0.17em}=40$ m, ∠$ACB={30}^o$ and ∠$ADB={60}^o$
If $AB=h$ m and $BC=x$ m, then $BD=(BC-CD)=(x-40)$ m.

Then, in the right ∆ABC, we have:
$\frac{AB}{BC}=\tan {30}^o=\frac{1}{\sqrt{3}}$
⇒ $\frac{h}{x}=\frac{1}{\sqrt{3}}$
⇒ $x=h\sqrt{3}$
In the right ∆ABD, we have:
$\frac{AB}{BD}=\tan {60}^o=\sqrt{3}$
⇒ $\frac{h}{(x-40)}=\sqrt{3}$
⇒ $h=\sqrt{3}(x-40)$
⇒ $h=\sqrt{3}(h\sqrt{3}-40)$ [∵ $x=h\sqrt{3}$ ]
⇒ $h=3h-40\sqrt{3}$ ⇒ $2h=40\sqrt{3}$ ⇒ $h=\frac{40\sqrt{3}}{2}=20\sqrt{3}$ m
Hence, assertion (A) is true.

Assertion (A):
Let:
$a=40$ m, $\alpha ={30}^o$ and $\beta ={60}^o$
Given:
$h=\frac{a\tan \alpha \tan \beta }{\tan \beta -\tan \alpha }$
Putting the values of $a,\alpha$ and $\beta$ in the above equation, we get:
$h=\frac{40\tan {30}^o\tan {60}^o}{\tan {60}^o-\tan {30}^o}=\frac{40\times {\displaystyle \frac{1}{\sqrt{3}}}\times \sqrt{3}}{\sqrt{3}-{\displaystyle \frac{1}{\sqrt{3}}}}=\frac{40\sqrt{3}}{2}=20\sqrt{3}$m
Hence, reason (R) is true.

Both assertion (A) and reason (R) are true, and (R) is the correct reason of assertion (A).