Question 1

1. If the length of the shadow of a vertical pole on the horizontal ground is equal to its height, then what is the angle of elevation of the sun?

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

Answer

✓ Correct Answer: Option B

Question 2

2. A ladder leaning against a wall makes an angle of $60^\circ$ with the horizontal ground. If the foot of the ladder is 2.5 m away from the wall, find the length of the ladder.

A) 5 mB) 2.5 mC) $5\sqrt{3} m$D) $\frac{5}{\sqrt{3}} m$

Question 3

3. The height of a tower is 10 m. What is the length of its shadow when the Sun's altitude is $30^\circ$?

A) $\frac{10}{\sqrt{3}} m$B) $10\sqrt{3} m$C) 10 mD) $5\sqrt{3} m$

Answer

✓ Correct Answer: Option B

Question 4

4. 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, making an angle of $60^\circ$ with the ground. Assuming there is no slack, what is the length of the string?

A) $40\sqrt{3} m$B) $60\sqrt{3} m$C) 120 mD) $20\sqrt{3} m$

Answer

✓ Correct Answer: Option A

Question 5

5. The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of the tower, is $30^\circ$. The height of the tower is:

A) $30\sqrt{3} m$B) $15\sqrt{3} m$C) $10\sqrt{3} m$D) 10 m

Answer

✓ Correct Answer: Option C

Question 6

6. An observer 1.5 m tall is 28.5 m away from a chimney. The angle of elevation of the top of the chimney from her eyes is $45^\circ$. What is the height of the chimney?

A) 27 mB) 28.5 mC) 30 mD) 31.5 m

Answer

✓ Correct Answer: Option C

Question 7

7. Two poles of equal heights are standing opposite each other on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are $60^\circ$ and $30^\circ$. The height of the poles is:

A) $10\sqrt{3} m$B) $20\sqrt{3} m$C) $40\sqrt{3} m$D) 20 m

Answer

✓ Correct Answer: Option B

Question 8

8. From the top of a 7 m high building, the angle of elevation of the top of a cable tower is $60^\circ$ and the angle of depression of its foot is $45^\circ$. Determine the height of the tower.

A) $7(\sqrt{3} - 1) m$B) $7\sqrt{3} m$C) 14 mD) $7(\sqrt{3} + 1) m$

Answer

✓ Correct Answer: Option D

Question 9

9. 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 of $30^\circ$ with it. The distance between the foot of the tree to the point where the top touches the ground is 8 m. Find the original total height of the tree.

A) $8\sqrt{3} m$B) $24\sqrt{3} m$C) $\frac{8}{\sqrt{3}} m$D) $16\sqrt{3} m$

Answer

✓ Correct Answer: Option A

Question 10

10. The angles of elevation of the top of a tower from two points at a distance of a and b from the base and in the same straight line with it are complementary. What is the height of the tower?

A) a + bB) $\sqrt{ab}$C) $\frac{a+b}{2}$D) ab

Answer

✓ Correct Answer: Option B

Question 11

11. The angle of depression of a car standing on the ground, from the top of a 15 m high tower, is $30^\circ$. The distance of the car from the base of the tower is:

A) $15\sqrt{3} m$B) $5\sqrt{3} m$C) 15 mD) 30 m

Answer

✓ Correct Answer: Option A

Question 12

12. An observer on top of a 100 m tall lighthouse sees two ships on the same side of the lighthouse. The angles of depression of the two ships are $45^\circ$ and $30^\circ$. What is the distance between the two ships?

A) $100(\sqrt{3} + 1) m$B) $100(\sqrt{3} - 1) m$C) 100 mD) $50\sqrt{3} m$

Answer

✓ Correct Answer: Option B

Question 13

13. The shadow of a tower standing on a level ground is found to be 40 m longer when the Sun's altitude drops from $60^\circ$ to $30^\circ$. The height of the tower is:

A) $40\sqrt{3} m$B) $20\sqrt{3} m$C) 20 mD) 60 m

Answer

✓ Correct Answer: Option B

Question 14

14. If the ratio of the length of a vertical rod and the length of its shadow is $1:\sqrt{3}$, then the angle of elevation of the sun is:

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

Answer

✓ Correct Answer: Option A

Question 15

15. An aeroplane is flying horizontally at a height of 3000 m. It passes vertically above another plane at an instant when the angles of elevation of the two planes from the same point on the ground are $60^\circ$ and $45^\circ$ respectively. The vertical distance between the aeroplanes at that instant is:

A) $1000(3 - \sqrt{3}) m$B) $1000(\sqrt{3} - 1) m$C) $3000(3 - \sqrt{3}) m$D) $1000\sqrt{3} m$

Answer

✓ Correct Answer: Option A

Question 16

16. A man on the top of a vertical tower observes a car moving at a uniform speed coming directly towards it. If it takes 6 seconds for the angle of depression to change from $30^\circ$ to $60^\circ$, how soon after this will the car reach the foot of the tower?

A) 2 secondsB) 3 secondsC) 4 secondsD) 6 seconds

Answer

✓ Correct Answer: Option B

Question 17

17. The angle of elevation of a stationary cloud from a point h meters above a lake is $\alpha$ and the angle of depression of its reflection in the lake is $\beta$. The height of the cloud above the lake surface is:

A) $h \left( \frac{\tan \beta - \tan \alpha}{\tan \beta + \tan \alpha} \right)$B) $h \left( \frac{\tan \alpha + \tan \beta}{\tan \alpha - \tan \beta} \right)$C) $h \left( \frac{\tan \beta + \tan \alpha}{\tan \beta - \tan \alpha} \right)$D) $h \left( \frac{\tan \alpha - \tan \beta}{\tan \alpha + \tan \beta} \right)$

Answer

✓ Correct Answer: Option C

Question 18

18. The tops of two poles of heights 16 m and 10 m are connected by a taut wire. If the wire makes an angle of $30^\circ$ with the horizontal, find the length of the wire.

A) 10 mB) 12 mC) 14 mD) 16 m

Answer

✓ Correct Answer: Option B

Question 19

19. A weather balloon is connected to a meteorological station by a cable of length 215 m inclined at $60^\circ$ to the ground. Determine the height of the balloon from the ground. (Assume the cable has no slack)

A) 107.5 mB) $215\sqrt{3} m$C) $107.5\sqrt{3} m$D) $\frac{215}{\sqrt{3}} m$

Answer

✓ Correct Answer: Option C

Question 20

20. From a point 20 m away from the base of a tower, the angle of elevation to the top of the tower is $45^\circ$. What is the height of the tower?

A) 10 mB) $20\sqrt{3} m$C) $\frac{20}{\sqrt{3}} m$D) 20 m

Answer

✓ Correct Answer: Option D

Grade 10 Math Quiz: Applications of Trigonometry

1. If the length of the shadow of a vertical pole on the horizontal ground is equal to its height, then what is the angle of elevation of the sun?

2. A ladder leaning against a wall makes an angle of $60^\circ$ with the horizontal ground. If the foot of the ladder is 2.5 m away from the wall, find the length of the ladder.

3. The height of a tower is 10 m. What is the length of its shadow when the Sun's altitude is $30^\circ$?

4. 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, making an angle of $60^\circ$ with the ground. Assuming there is no slack, what is the length of the string?

5. The angle of elevation of the top of a tower from a point on the ground, which is 30 m away from the foot of the tower, is $30^\circ$. The height of the tower is:

6. An observer 1.5 m tall is 28.5 m away from a chimney. The angle of elevation of the top of the chimney from her eyes is $45^\circ$. What is the height of the chimney?

7. Two poles of equal heights are standing opposite each other on either side of the road, which is 80 m wide. From a point between them on the road, the angles of elevation of the top of the poles are $60^\circ$ and $30^\circ$. The height of the poles is:

8. From the top of a 7 m high building, the angle of elevation of the top of a cable tower is $60^\circ$ and the angle of depression of its foot is $45^\circ$. Determine the height of the tower.

9. 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 of $30^\circ$ with it. The distance between the foot of the tree to the point where the top touches the ground is 8 m. Find the original total height of the tree.

10. The angles of elevation of the top of a tower from two points at a distance of a and b from the base and in the same straight line with it are complementary. What is the height of the tower?

11. The angle of depression of a car standing on the ground, from the top of a 15 m high tower, is $30^\circ$. The distance of the car from the base of the tower is:

12. An observer on top of a 100 m tall lighthouse sees two ships on the same side of the lighthouse. The angles of depression of the two ships are $45^\circ$ and $30^\circ$. What is the distance between the two ships?

13. The shadow of a tower standing on a level ground is found to be 40 m longer when the Sun's altitude drops from $60^\circ$ to $30^\circ$. The height of the tower is:

14. If the ratio of the length of a vertical rod and the length of its shadow is $1:\sqrt{3}$, then the angle of elevation of the sun is:

16. A man on the top of a vertical tower observes a car moving at a uniform speed coming directly towards it. If it takes 6 seconds for the angle of depression to change from $30^\circ$ to $60^\circ$, how soon after this will the car reach the foot of the tower?

17. The angle of elevation of a stationary cloud from a point h meters above a lake is $\alpha$ and the angle of depression of its reflection in the lake is $\beta$. The height of the cloud above the lake surface is:

18. The tops of two poles of heights 16 m and 10 m are connected by a taut wire. If the wire makes an angle of $30^\circ$ with the horizontal, find the length of the wire.

19. A weather balloon is connected to a meteorological station by a cable of length 215 m inclined at $60^\circ$ to the ground. Determine the height of the balloon from the ground. (Assume the cable has no slack)

20. From a point 20 m away from the base of a tower, the angle of elevation to the top of the tower is $45^\circ$. What is the height of the tower?