Answer:
It is given that The top of a tree makes angles s and t with Points K and L on the ground, respectively, such that the angles are complementary which means that s+t=90°.
Now, from the figure,
\frac{AB}{BC}=tans
⇒\frac{h}{x}=tans (1)
Also, \frac{AB}{BD}=tant
⇒\frac{h}{y}=tant (2)
Now, multiply equation (1) and (2), we get
\frac{h}{x}{\times}\frac{h}{y}=tans{\times}tant
Since, s and t are complementary, therefore
\frac{h}{x}{\times}\frac{h}{y}=1
⇒h^2=xy
⇒h=\sqrt{xy}
Therefore, the height of the tree in terms of x and y is: h=\sqrt{xy}
Now, s=38°⇒s+t=90°⇒t=90-38=52°
Thus, t=52°
Now, substitute the value of y=3 and t=52° in equation (2), we get
\frac{h}{3}=tan52^{\circ}
⇒h=3(1.279)
⇒h=3.84 meters