CLASS-10

BOARD: CBSE

Mathematic Worksheet - 7

TOPIC: Heights and Distance

For other **CBSE Worksheet for class 10 Mathematic** check out main page of Entrancei.

**SUMMARY**

**ANGLE OF ELEVATION**

The line joining the object and eye of the observer is known as the line of sight and the angle which this line of sight makes with the horizontal drawn through the eye of the observer is known as the **angle of elevation**.

**ANGLE OF DEPRESSION**

When the object is at a lower level than the observer’s eyes, he has to look downwards to have a view of the object. In that case, the angle which the line of sight makes with the horizontal drawn through the observer’s eye is known as the **angle of depression**.

**OBJECTIVE**

**1. The angle of elevation of the top of a tower as observed from a point on the horizontal ground is ‘x’. If we move a distance ’d’ towards the foot of the tower, the angle of elevation increases to ‘y’, then the height of the tower is:**

A. (d tanx tany )/tanx-tany

(b) d(tanx+tany )

(c) d( tanx-tany )

(d) (d tanx tany )/tanx+tany -

**2. The angle of elevation of the top of a tower, as seen from two points A and B situated in the same line and at distances ‘p’ and ‘q’ respectively from the foot of the tower, are complementary, then the height of the tower is:**

A. Pq

(b) P/q

(c) √pq

(d) none of these

**3. The angle of elevation of the top of a tower at a distance of metres from the foot is 60 ^{0}. Find the height of the tower.**

(a)metres

(b) metres

(c) – 50 metres

(d) 50 metres

**4. The shadow of a tower, when the angle of the sun is 30 ^{0}, is found to be 5 m longer than when it was 45^{0}, then the height of the tower in meters is:**

A. 5/(√3+1)

(b) 5/2 (√3-1)

(c) 5/2 (√3+1)

(d) none of these

**SUBJECTIVE**

**1. From the top of a light house, the angles of depression of two ships on the opposite sides of it are observed to be α and β. If the height of the light house be h metres and the line joining the ships passes through the foot of the light house. Show that the distance between the ships is (h (tanα+tanβ) )/tanα tanβ meters.**

**2. A ladder rests against a wall at angle α to the horizontal. Its foot is pulled away from the previous point through a distance ‘a’, so that it slides down a distance ‘b’ on the wall making an angle β with the horizontal. Show that a/b = .cosα - cosβ /sinβ - sinα**

**3. From an aero plane vertically above a straight horizontal road, the angle of depression of two consecutive kilometer stones on opposite sides of aeroplane are observed to be α and β. Show that the height of the aeroplane above the road is tan α tanβ /tanα +tanβ kilometer.**

**4. A round balloon of radius ‘r’ subtends an angle θ at the eye of an observer while the angle of elevation of its centre is φ. Prove that the height of the centre of the balloon is rsinφ cosec θ/2.**

**5. A man on a cliff observes a boat at an angle of depression of 30 ^{0} which is approaching the shore to the point immediately beneath the observer with a uniform speed. Six minutes later, the angle of depression of the boat is found to be 60^{0}. Find the total time taken by the boat from the initial point to reach the shore.**

**6. At a point, the angle of elevation of a tower is such that its tangent is 5/12. On walking 240 m near the tower, the tangent of the angle of elevation becomes 3/4. Find the height of the tower.**

**7. From a window ‘x’ meters high above the ground in a street, the angles of elevation and depression of the top and foot of the other house on the opposite side of the street are α and β respectively. Show that the opposite house is x(1 + tanα cotβ ) meters.**

**8. From the top of a building 60 m high, the angles of depression of the top and bottom of a vertical lamp post are observed to be 30 ^{0} and 60^{0} respectively. Find**

1. The horizontal distance between the building and the lamp post.

2. The height of the lamp post. [Take √3 = 1.732]

**9. An aero plane when flying at a height of 3125 m from the ground passes vertically below another plane at an instant when the angles of elevation of the two planes from the same point on the ground are 30 ^{0} and 60^{0} respectively. Find the distance between the two planes at that instant.**

**10. The angle of elevation of a cloud from a point 60 m above a lake is 30 ^{0} and the angle of depression of the reflection of the cloud in the lake is 60^{0}. Find the height of the cloud from the surface of the lake.**

**Answers: **

**Objective: **

1) a

2) c

3) d

4) c

**Subjective: **

5) 9 min

6) 225m

7) ?

8) Horizontal distance between the building and the lamp post=34.64 m and height of the lamppost =40 m

9) 6250m

10)120 m