Calculating Heights and Distances

Q.

Question 15
A pole 6m high casts a shadow $2\sqrt{3}m$ long on the ground, then the Sun’s elevation is

(A) ${60}^{\circ }$
(B) ${45}^{\circ }$
(C) ${30}^{\circ }$
(D) ${90}^{\circ }$

Q.

Question 16
The angles of elevation of the top of a tower from two points at a distance of 4 m and 9 m from the base of the tower and in the same straight line with it are complementary. Prove that the height of the tower is 6 m.

Q.

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

1. 30 m

2. 30√3 m

3. 20 m

4. 20 m

Q.

Find the integral of $\int \frac{\cos ecx}{\log \left(\tan \frac{x}{2}\right)}dx$.

Q.

A vertical stick 10cm long casts a shadow 8cm long. At the same time, a tower casts a shadow 30m long. Determine the height of the tower?

1. 65m

2. 75m

3. 62.5m

4. 100m

Q.

A circus artist is climbing a 30 m long rope, which is tightly stretched and tied from the top of a vertical pole to the ground. Find the distance of the pole to the peg in the ground, if the angle made by the rope with the ground level is ${30}^{\circ }$.

1. 15$\sqrt{3}$ m

2. 5 m

3. 20 m

4. 18 m

Q.

cos A is the abbreviation used for the cosecant of angle A.

1. True

2. False

Q. Sin( 90 - theta ) = cos theta prove this using coordinate axes

Q. A vertical pole fixed to the ground is divided in the ratio 1 $:$ 9 by a mark on it with the lower part shorter than the upper part. If the two parts subtend equal angles at a place on the ground, $15m$ away from the base of the pole, what is the height of the pole?

1. $60\sqrt{5}m$
2. $15\sqrt{5}m$
3. $15\sqrt{3}m$
4. $60\sqrt{3}m$

Q. In a triangle ABC, cos[(A+B)/2)] is equal to

1. cos C/2
2. sin C/2
3. Sec C/2
4. tan C/2

Q. Question 6
The angle of elevation of the top of a tower from two points distant s and t from its foot are complementary. Prove that the height of the tower is $\sqrt{st}$

Q.

An observer $\sqrt{3}$m tall is 3 m away from the pole 2$\sqrt{3}$m high. What is the angle of elevation of the top? [2 MARKS]

Q.

A kite is flying on a string of length 40$\sqrt{3}$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 ${60}^{\circ }$. Find the height at which the kite is flying.

1. 60√3 m

2. 40 m

3. 20 m

4. 60 m

Q.

In ΔABC, right angled at B, if cot A = √3 then the value of cos A x sin C + sin A x cos C =

1. 

2. 

3. 

4. 

Q. If xsin(90^° - theta) cot(90^°-theta) = cos(90^°-theta), then x is equal to 1. 0 2. 1 3. -1 3. 2

Q. A flagstaff is placed on top of a building. The flagstaff and the building subtend equal angles at a point on level ground which is $200m$ away from the foot of the building. If the height of the flagstaff is $50m$ and the height of the building is $h$, which of the following is true?

1. $h^3-50h^2+(200{)}^{2}h+50(200{)}^2=0$
2. $h^3+50h^2+(200{)}^{2}h-50(200{)}^2=0$
3. $h^3-50h^2-(200{)}^{2}h+50(200{)}^2=0$
4. none of these

Q.

The angles of elevation of the top of a tower standing on a horizontal plane from two points on a line passing through the foot of the tower at distances $49m$ and $36m$ are ${43}^{\circ }$ and ${47}^{\circ }$ respectively. What is the height of the tower?

1. $40m$

2. $42m$

3. $45m$

4. $47m$

Q. For all values of A, B and C, prove that:- cosA + cosB + cosC + cos (A+B+C)= 4cos (A + B/2). cos (B+C/2). cos (C+ A/2)

Q. If cosec x = (13/12), find the value of tan x.

1. 12/13
2. 5/12
3. 13/5
4. 12/5

Q. Question 7
From a point on the ground, the angles of elevation of the bottom and the top of a transmission tower fixed at the top of a 20 m high building are ${45}^{\circ }$ and ${60}^{\circ }$ respectively. Find the height of the tower.

Q. A tree 12 m high, is broken by the wind in such a way that its top touches the ground and makes an angle ${60}^{\circ }$ with the ground. At what height from the bottom the tree is broken by the wind? [3 MARKS]

Q.

The angles of depression of the top and the bottom of a 10 m tall building from the top of a multi-storeyed building are ${30}^{\circ }$ and ${45}^{\circ }$, respectively. Find the height of the multi-storeyed building.

1. 10 m

2. 15 m

3. 5 m

4. 5($\sqrt{3}$ + 3) m

Q.

If $A$ and $B$ are complementary angles, $\cot A\cdot \tan B$ is equal to?

Q.

Two towers A and B are standing at some distance apart. From the top of tower A, the angle of depression of the foot of tower B is found to be ${30}^{\circ }$. From the top of tower B, the angle of depression of the foot of tower A is found to be ${60}^{\circ }$. If the height of tower B is ‘h’ m then the height of tower A in terms of ‘h’ is _____ m

1. $\frac{h}{2}$m

2. $\frac{h}{3}$m

3. $\sqrt{3}h$m

4. $\frac{h}{\sqrt{3}}$m

Q.

From a point on a bridge across a river, the angles of depression of the banks on opposite sides of the river are ${30}^{\circ }$ and ${45}^{\circ }$, respectively. If the width of the river is 12 m , find the height at which the bridge is

1. 6(√3 -1) m

2. 6√3 m

3. 6 m

4. 3 m

Q.

cos θ/(1 - tan θ) + sin θ/(1 - cot θ) = sin θ + cos θ

Q.

If the angle of elevation of Sun decreases from ${90}^{\circ }$ to $0^{\circ }$, then the length of the shadow of a man will :

1. Decreases

2. Doesn't change

3. Increases

4. Can't be determined

Q. If a man has a ladder of 18 m and needs to reach the top of the wall of $9\sqrt{3}m$. What is the required angle between ladder and wall?

1. ${30}^{\circ }$

2. ${60}^{\circ }$

3. ${45}^{\circ }$

4. ${15}^{\circ }$

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 his line of sight of the tip of the other building to be ${30}^{\circ }$. Based on the information in the question, find the height of the adjacent building?

1. 10√3feet

2. 55+10√3 feet

3. 55feet

4. 60cm

Q.

A bridge is made over the river which has constant width throughout. The bridge makes an angle of ${30}^{\circ }$ from the direction of flow (or makes an angle of ${60}^{\circ }$ from a line perpendicular to the width of the river). If the length of the bridge is 100 m, then find the width of the river.

1. 30 m
2. 50 m
3. 100 m
4. 120 m