[650 m] From the foot and the top of a building of height 230 m, a person observes the top of a tower with a

From the foot and the top of a building of height 230 m, a person observes the top of a tower with angles of elevation of b and a respectively. What is the distance between the top of these buildings if tan a = 5/12 and tan b = 4/5

400 m

250 m

600 m

650 m

Explanation:

Let ED be the building and AC be the tower.

Given that ED = 230 m, $\angle$ADC = b, $\angle$AEB = a
Also given that tan a = 5/12 and tan b = 4/5

Let AC = h

Required Distance = Distance between the top of these buildings = AE

From the right $\triangle$ ABE,
tan(a)=$\dfrac{AB}{BE}$
=> $\dfrac{5}{12}=\dfrac{(h-230)}{BE}$ [∵ tan(a)=5/12(given), AB = (AC-BC) = (AC-ED) = (h-230)]
=> BE =$\dfrac{12(h-230)}{5}$ ⋯(eq:1)
From the right $\triangle$ ACD,
tan(b)=$\dfrac{AC}{CD}$

=> $\dfrac{4}{5}=\dfrac{h}{CD}$ [∵ tan(b) = 4/5(given), AC=h]

=> CD =$\dfrac{5}{h4}$ ⋯(eq:2)

From the diagram, BE = CD
⇒$\dfrac{12(h-230)}{5}=\dfrac{5h}{4}$ (from eq:1 & eq:2)
⇒48h−(4×12×230)=25h
⇒23h=(4×12×230)

⇒h=$\dfrac{(4×12×230)}{23}$=480 m ⋯(eq:3)

AB = (AC - BC)
= (480 - 230) [∵ Since AC=h=480(from eq:3) and BC=ED=230 m(given)]
= 250 m

In the triangle ABE, tan(a) = 5/12. Let s figure out the value of sin(a) now.

Consider a triangle with opposite side = 5 and adjacent side = 12 such that tan(a) = 5/12
hypotenuse =$\sqrt{5^{2}+12^{2}}$=13

i.e., sin(a) =$\dfrac{opposite}{ sidehypotenuse}=\dfrac{5}{13}$
We have seen that sin(a) = $\dfrac{5}{13}$

⇒$\dfrac{AB}{AE}=\dfrac{5}{13}$
⇒AE = AB×$\dfrac{13}{5}$
=250×$\dfrac{13}{5}$=650 m

i.e., Distance between the top of the buildings = 650 m

From the top of a hill 100 m high, the angles of depression of the top and bottom of a pole are 30° and 60° respectively. What is the height of the pole?	Answer
A flagstaff is placed on top of a building. The flagstaff and building subtend equal angles at a point on level ground which is 200 m away from the foot of the building. If the height of the flagstaff is 50 m and the height of the building is h, which of the following is true?	Answer
Two vertical poles are 200 m apart and the height of one is double that of the other. From the middle point of the line joining their feet, an observer finds the angular elevations of their tops to be complementary. Find the heights of the poles.	Answer
When the sun s altitude changes from 30° to 60°, the length of the shadow of a tower decreases by 70m. What is the height of the tower?	Answer
A person, standing exactly midway between two towers, observes the top of the two towers at angle of elevation of 22.5° and 67.5°. What is the ratio of the height of the taller tower to the height of the shorter tower? (Given that tan 22.5° = √2−1)	Answer
21. The elevation of the summit of a mountain from its foot is 45°. After ascending 2 km towards the mountain upon an incline of 30°,the elevation changes to 60°. What is the approximate height of the mountain?	Answer
On the same side of a tower, two objects are located. Observed from the top of the tower, their angles of depression are 45° and 60°. If the height of the tower is 600 m, the distance between the objects is approximately equal to :	Answer
A man on the top of a vertical observation tower observers a car moving at a uniform speed coming directly towards it. If it takes 8 minutes for the angle of depression to change from 30° to 45°, how soon after this will the car reach the observation tower?	Answer
The angle of elevation of the top of the tower from a point on the ground is sin⁻¹$\left(\dfrac{3}{5}\right)$. If the point of observation is 20 meters away from the foot of the tower, what is the height of the tower?	Answer
From the foot and the top of a building of height 230 m, a person observes the top of a tower with angles of elevation of b and a respectively. What is the distance between the top of these buildings if tan a = 5/12 and tan b = 4/5	Answer