Height and Distances

Q.

The value of $\frac{cot54°}{\tan 36°}+\frac{\tan 20°}{cot70°}$ is

1. $2$

2. $3$

3. $1$

4. $0$

Q. A balloon is connected to a meteorological ground station by a cable of length, 215 m inclined at 60° to the horizontal. Determine the height of the balloon from the ground. Assume that there is no slack in the cable.
[Consider $\sqrt{3}=1.73$]

1. 
   150 m
2. 
   145.8 m
3. 
   137.6 m
4. 
   185.975 m

Q.

From the top of a cliff $50$m high, the angles of depression of the top and bottom of a tower are observed to be $30°$ and $45°$.

The height of tower is

1. $50$m

2. $50\sqrt{3}$m

3. $50(\sqrt{3}–1)$m

4. $50(1–\frac{\sqrt{3}}{3})$m

Q.

A 6 feet tall man sees an apple on the ground, $6\sqrt{3}$ feet away from him. What is the angle of depression when he is looking at the apple?

1. $\text{can't be determined}$

2. ${30}^{\circ }$

3. ${60}^{\circ }$

4. ${45}^{\circ }$

Q.

Question 2
A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle 30° with it. The distance between the foot of the tree to the point where the top touches the ground is 8 m. Find the height of the tree.

Q.

In the given right angled triangle ABC, if $\alpha ={30}^{\circ }$ and $AC=10cm$ then the side $BC$ would be ___ $cm$

Q. Find the approximate length of AB from the given image. (Take $\sqrt{3}=1.73$)

1. 26 m
2. 27 m
3. 30 m
4. 25 m

Q.

The height of a tree is 10$\sqrt{3}$ m, if a boy looks at the top of the tree with an angle of elevation of ${30}^{\circ }$, find the distance between the boy and the tree.

1. 10$\sqrt{3}$ m

2. 10 m

3. 30 m

4. 20 m

Q. Question 7
The shadow of a tower standing on a level plane is found to be 50 m longer when Sun’s elevation is ${30}^{\circ }$ than when it is ${60}^{\circ }$. Find the height of the tower.

Q. Question 5
A kite is flying at a height of 60 m above the ground. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is ${60}^{\circ }$. Find the length of the string, assuming that there is no slack in the string.

Q. An aeroplane at an altitude of 250m observes the angle of depression of two boats on opposite banks of a river to be ${45}^o$ and ${60}^o$ respectively. Find width of the river.
​​​​​​​

1. $250(1-\frac{1}{\sqrt{3}})m$
2. $250(\sqrt{3}+\frac{1}{\sqrt{3}})m$
3. $250(\sqrt{3}-\frac{1}{\sqrt{3}})m$
4. $250(1+\frac{1}{\sqrt{3}})m$

Q.
The height of the building is 7 m, the height of the tower is.

1. $(7+7\sqrt{3})m$
2. $(7-7\sqrt{3})m$
3. $\sqrt{3}m$

Q. Angle of depression is the angle between the horizontal plane and the line from the observer’s eye to some object above his eye.

1. False
2. True

Q.

A tree breaks due to a storm and the broken part bends so that the top of the tree touches the ground making an angle ${30}^{\circ }$ with it. The distance between the foot of the tree to the point where the top touches the ground is 8m. Find the height of the tree?

1. $15\sqrt{3}m$

2. $20\sqrt{3}m$

3. $11\sqrt{3}m$

4. $8\sqrt{3}m$

Q. The angles of elevation of the top of a tower from two points on the ground at distances a metres and b metres from the base of the tower and in the same straight line are complemetary. Prove that the height of the tower is $\sqrt{ab}$ metres.

Q.

Two pillars of equal heights stand on either side of a road which is 100 m wide. At a point on the road between the pillars, the angles of elevation of the tops of the pillars are ${60}^{\circ }$ and ${30}^{\circ }$. Find the height of each pillar and position of the point on the road. [Take $\sqrt{3}$=1.732] [3 MARKS]

Q.

Sam is looking at the roof of the adjacent building from top of his flat. His height is $5$ feet. The height of his building as well as the distance between the two buildings is $50$ feet. Sam measures the angle of elevation of his line of sight to the top of the other building to be ${30}^{\circ }$. Based on the information in the question, find the height of the adjacent building?

1. 60cm

2. 55feet

3. 10√3feet

4. 55+10√3 feet

Q.

Sam is looking at the roof of the adjacent building from top of his flat. His height is 5 feet. He knows that the height of his building and the distance between the two buildings is 50 feet. Sam measures the angle of elevation of the tip of other building to be ${30}^{\circ }$. Based on the information in the question, find the height of the adjacent building?

1. 55feet

2. 10√3feet

3. 55+10√3 feet

4. 60cm

Q.
ABCD is a rectangle such that $\angle CAB={60}^{\circ }$ and $AC=aunits$.
The area of rectangle ABCD is ______.

1. $\frac{\sqrt{3}}{2}{a}^2$
2. $\frac{\sqrt{2}}{3}{a}^2$
3. $\frac{\sqrt{3}}{4}{a}^2$
4. $\frac{\sqrt{3}}{8}{a}^2$

Q. A ladder rests against a wall so that its top touches the roof of the house. If the ladder makes an angle of ${60}^{\circ }$ with the horizontal and height of the house be $6\sqrt{3}\text{m}$, then the length of the ladder is:

1. $12\sqrt{3}\text{m}$
2. $12\text{m}$
3. $\frac{12}{\sqrt{3}}\text{m}$
4. None of these

Q. Find the image of the point (3, 8) with respect to the line $x+3y=7$ assuming the line to be a plane mirror.

Q.

A kite is attached to a string of length 20$\sqrt{3}$m is tied to a pole on the level ground. If the string of the kite makes an angle of elevation of ${60}^{\circ }$ with the ground, then the kite is flying at a height of 20 m.

1. False

2. True

Q. lf the angle of elevation of the top of a tower from a point is ${60}^o$ and 40 metres vertically above this point the angle of elevation is ${45}^o$ The height of the tower in metres is

1. $64.64$
2. $94.64$
3. $54.64$
4. $74.64$

Q. On a horizontal plane, there is a vertical tower with a flagpole on the top of the tower. At a point $9$ meters away from the foot of the tower the angle of elevation of the top and bottom of the flagpole are ${60}^o$ and ${30}^o$ respectively. Find the height of the tower and the flagpole mounted on it.

Q.

A tree breaks due to storm and the broken part bends so that the top of the tree touches the ground making an angle ${30}^{\circ }$ with it. The distance between the foot of the tree to the point where the top touches the ground is 12 m. Find the height of the tree.

1. 12√3 m

2. 4√3 m

3. 12 m

4. 4 m

Q. ABC is an isosceles right angle triangle, right angled at B. The ratio of the sides AB: BC : AC is .

1. $1:1:\sqrt{2}$
2. $\sqrt{2}:1:1$
3. $\sqrt{3}:2:1$
4. $1:2:\sqrt{3}$

Q. In right angle $△ABC,\angle B={90}^o,\angle A={45}^o$, then $\tan {45}^o=$

1. $1$
2. $\sqrt{2}$
3. $\frac{1}{\sqrt{2}}$
4. None of above

Q. If $4{\cos }^2{x}^0-1=0$ and $0\le x^0\le {90}^0$, find $\frac{1}{{\cos }^2{x}^0}-{\tan }^2{x}^0$

Q.

Two poles of different heights are erected from the ground. If the smaller pole is $20m$ high and distance between two poles are $10m,$ and the angle of elevation of the top of the longer pole from that of the shorter pole is ${60}^{\circ }$, find the length of the second pole.

1. $20+\frac{10}{\sqrt{3}}m$

2. $20+5\sqrt{3}m$
   

3. $10+10\sqrt{3}m$

4. $20+10\sqrt{3}m$

Q.

A kite is flying on a string of 40 m. The string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is ${30}^{\circ }$. Find the height at which the kite is flying.

1. 20 m

2. 40 m

3. 10 m

4. 30 m