Some Applications of Trigonometry: Class 10 Important Questions with Answers

HOTS (higher-order) practice

H1. The angles of elevation of the top of a tower from two points at distances 4 m and 9 m from its base, on the same side and in a straight line, are complementary. Prove that the height of the tower is 6 m.

H2. From a window 15 m above the ground, the angles of elevation and depression of the top and the foot of a house opposite are 30° and 45°. Find the height of the house.

H3. A ladder leans against a wall at 60° with the ground; its foot is 2.5 m from the wall. Find the length of the ladder and how high it reaches.

H4. The shadow of a tower is 40 m longer when the sun's altitude is 30° than when it is 60°. Find the height of the tower.

Case-based practice

20 MCQs (1 mark each)

1. A tower 50 m high casts a 50 m shadow. The sun's altitude is
   (a) 30° (b) 45° (c) 60° (d) 90°
2. A ladder 10 m long reaches a window 5 m high. Its angle with the ground is
   (a) 30° (b) 45° (c) 60° (d) 90°
3. From 20 m from a tree's foot the elevation of its top is 60°. The tree is
   (a) $20\sqrt{3}$ m (b) $\frac{20}{\sqrt{3}}$ m (c) 20 m (d) 40 m
4. If the elevation of the sun increases, the shadow of a pole
   (a) increases (b) decreases (c) stays the same (d) doubles
5. A kite string 100 m long makes 30° with the ground. The kite's height is
   (a) 50 m (b) $50\sqrt{3}$ m (c) 100 m (d) $100\sqrt{3}$ m
6. The angle of depression of a car from a 75 m tower is 30°. The car is at
   (a) 75 m (b) $75\sqrt{3}$ m (c) 25√3 m (d) 150 m
7. If the shadow is $\sqrt{3}$ times the height, the elevation is
   (a) 30° (b) 45° (c) 60° (d) 15°
8. The angle of elevation of the top of a 15 m pole from a point 15 m away is
   (a) 30° (b) 45° (c) 60° (d) 75°
9. The angle of depression equals the angle of elevation because they are
   (a) vertically opposite (b) alternate (c) corresponding (d) complementary
10. A pole casts a shadow of $2\sqrt{3}$ m when the elevation is 60°. Its height is
    (a) 2 m (b) 6 m (c) $6\sqrt{3}$ m (d) 3 m
11. A 6 m pole's shadow is 6 m. The angle of elevation of the sun is
    (a) 30° (b) 60° (c) 45° (d) 90°
12. An observer 1.5 m tall is 28.5 m from a tower; elevation from the eye is 45°. Tower height
    (a) 28.5 m (b) 30 m (c) 27 m (d) 1.5 m
13. Elevation 30° from a point 100 m from a tower's foot. Height
    (a) 100 m (b) $100\sqrt{3}$ m (c) $\frac{100}{\sqrt{3}}$ m (d) 50 m
14. Two points on opposite sides of a 30 m tower see its top at 45° each. Distance between them
    (a) 30 m (b) 60 m (c) $30\sqrt{2}$ m (d) 15 m
15. If the angle of elevation of a cloud is 45° from a point 200 m away horizontally, its height is
    (a) 100 m (b) 200 m (c) $200\sqrt{2}$ m (d) 400 m
16. A wire tied from the top of a 12 m pole to the ground makes 60° with the ground. Its length
    (a) $8\sqrt{3}$ m (b) 24 m (c) $12\sqrt{3}$ m (d) 6 m
17. From a 40 m tower, the depression of a point is 45°. Its distance from the foot
    (a) 40 m (b) $40\sqrt{3}$ m (c) 20 m (d) 80 m
18. A boy moves towards a tower; elevation goes from 30° to 60°. The distance from the tower
    (a) becomes one third (b) halves (c) doubles (d) stays same
19. Line of sight to an object below the eye forms the
    (a) angle of elevation (b) angle of depression (c) right angle (d) straight angle
20. The height of a tower is 10 m. The length of its shadow when the sun's altitude is 30° is
    (a) 10 m (b) $10\sqrt{3}$ m (c) $\frac{10}{\sqrt{3}}$ m (d) 20 m

Answer key

1 (b) · 2 (a) · 3 (a) · 4 (b) · 5 (a) · 6 (b) · 7 (a) · 8 (b) · 9 (b) · 10 (b) · 11 (c) · 12 (b) · 13 (c) · 14 (b) · 15 (b) · 16 (a) · 17 (a) · 18 (a) · 19 (b) · 20 (b)

Hints: Q2 $\sin \theta =\frac{5}{10}$. Q10 $2\sqrt{3}\tan 60°=6$. Q12 28.5 + 1.5. Q16 $\frac{12}{\sin 60°}=8\sqrt{3}$. Q18 $h\sqrt{3}\to \frac{h}{\sqrt{3}}$: one third.

Assertion–Reason (10)

(a) Both A and R are true and R is the correct explanation of A. (b) Both A and R are true but R is not the correct explanation of A. (c) A is true but R is false. (d) A is false but R is true.

1. A: If the shadow of a tower equals its height, the sun's altitude is 45°. R: tan 45° = 1.
2. A: The angle of elevation of a tower's top from a point increases as the point moves away. R: tan θ increases with θ.
3. A: From the top of a cliff, the angle of depression of a boat equals the angle of elevation of the cliff top from the boat. R: Alternate angles between parallel lines are equal.
4. A: A 10 m ladder at 60° with the ground reaches 5 m up the wall. R: sin 60° = $\frac{\sqrt{3}}{2}$.
5. A: For two ships on opposite sides of a lighthouse, the distance between them is the sum of their distances from the foot. R: The foot lies between them on the line joining them.
6. A: The height of a kite 80 m along a string at 30° is 40 m. R: sin 30° = $\frac{1}{2}$.
7. A: As the sun rises higher, shadows get shorter. R: For fixed height, the shadow is $h\cot \theta$, which decreases as θ increases.
8. A: The angle of depression of an object on the ground can be 90° only if the object is directly below. R: Angle of depression is measured from the vertical.
9. A: A pole 6 m high with shadow $2\sqrt{3}$ m means sun's elevation 60°. R: $\tan 60°=\sqrt{3}$.
10. A: If the elevation of a tower's top from 4 m and 9 m (same side) are complementary, the tower is 6 m. R: $\tan \theta \cdot \tan (90°-\theta )=1$.

Key: 1 (a) · 2 (d) · 3 (a) · 4 (d) · 5 (a) · 6 (a) · 7 (a) · 8 (c) · 9 (a) · 10 (a)

Q2: moving away, the angle decreases ⇒ A false. Q4: height $=10\sin 60°=5\sqrt{3}$ ⇒ A false. Q8: depression is measured from the horizontal ⇒ R false.

Very short answer (1 mark each)

1. A tower 100 m high; elevation from a point is 45°. Distance of the point from the foot? Ans. 100 m
2. The shadow of a 15 m pole is $5\sqrt{3}$ m. Sun's elevation? Ans. 60°
3. A kite at 60 m height, string at 60°. String length? Ans. $40\sqrt{3}$ m
4. Define the angle of depression. Ans. Angle between the horizontal at the eye and the line of sight to an object below.
5. From a 20 m tower, depression of a car is 60°. Its distance? Ans. $\frac{20}{\sqrt{3}}$ m
6. A ladder makes 30° with a wall; foot is 2 m from the wall. Ladder length? Ans. 4 m
7. If height = $\frac{1}{\sqrt{3}}$ × shadow, elevation? Ans. 30°
8. Which ratio links height and slant length? Ans. sin θ

Short answer practice (3 marks each)

1. A vertical tower stands on level ground. From a point 20 m from its foot, the angle of elevation of the top is 60°. Find the height and the length of the line of sight to the top.

2. The angle of elevation of the top of a building from the foot of a tower is 30° and that of the top of the tower from the foot of the building is 60°. If the building is 15 m high, find the height of the tower.

3. A man on a ship 10 m above water sees the top of a hill at 60° elevation and its base at 30° depression. Find the distance of the hill and its height.

Long answer practice (5 marks each)

4. From the top of a 100 m lighthouse, the angles of depression of two boats on the same side are 30° and 45°. Find the distance between the boats.

5. A 1.5 m tall observer is 28.5 m from a chimney. The angle of elevation of its top from his eyes is 45°. Then he walks towards it until the elevation is 60°. Find the height of the chimney and the distance walked.

6. The angles of elevation of the top of a tower from two points P and Q at distances a and b from its base, on the same straight line and the same side, are complementary. Prove that the height is $\sqrt{ab}$.

7. An aeroplane at 3000 m flies horizontally. From a point on the ground, the angle of elevation changes from 60° to 45° in 15 s. Find its speed in km/h.