1. There is a flagpole on top of a monument. When the sun’s elevation angle is 30°, the shadow length of the flagpole is 3√3 meters. > What is the height of the flagpole? > If the height of the monument is 15 meters, then what will be the total shadow length of the monument including the flagpole?

2. When the sun's angle of elevation is 45°, the length of the shadow of a vertical pillar on a horizontal plane is a certain value. When the angle of elevation becomes 30°, the shadow length increases by 60 meters compared to the previous case. Find the height of the pillar.

3. When the angle of elevation of the sun is 45°, the length of the shadow of a vertical pillar is a certain value. When the angle of elevation decreases to 30°, the length of the shadow becomes 60 meters longer than before. Determine the height of the pillar.

4. If the length of a pillar's shadow is equal to its height, the angle of elevation of the sun will be-

(a) 30° (b) 60° (c) 45° (d) 90°

5. If the sun's elevation angle is _____, then the length of the pillar's shadow will be equal to the height of the pillar.

6. The English translation is: "When the sun's angle of elevation increases from 45° to 60°, the length of the shadow of a pole decreases by 3 meters. Determine the height of the pole. (Take \(\sqrt{3} = 1.732\))"

7. If the length of a shadow at the base of a post is √3 times the height of the post, then the angle of elevation of the Sun is:

(a) 30° (b) 45° (c) 60° (d) none of the above

8. When the angle of elevation of the Sun is 45°, the height of a post and the length of its shadow will be ________.

9. When the sun's altitude angle decreases from 60° to 30°, the length of the shadow of a vertical rod increases by 40 meters. What is the height of the rod?

(a) \(10\sqrt3\) meter (b) \(15\sqrt3\) meter (c) \(5\sqrt3\) meter (d) \(20\sqrt3\) meter

10. If the ratio of the length of a shadow of a pillar to the height of the pillar is √3:1, find the angle of elevation of the sun.

11. As the sun's angle of elevation increases from 30° to 60°, the length of the monument's shadow decreases by \(50\sqrt{3}\) meters. What is the height of the monument?

12. When the sun's elevation angle increases from 45° to 60°, the length of the shadow of a pole decreases by 3 meters. Find the height of the pole. [Take √3 = 1.732 and calculate the approximate value up to three decimal places.]

13. From a point 150 meters away from the base of an incomplete pillar, the angle of elevation to its top is 45°. How much additional height must be added to the pillar so that, from the same point, the angle of elevation to the new top becomes 60°?

14. When the sun’s angle of elevation changes from 45° to 60°, the shadow of a telegraph pole changes by 4 meters. Find the length of the shadow of the same pole when the angle of elevation is 30°.

15. When the sun's elevation angle increases from 45° to 60°, the length of a pole's shadow decreases by 3 meters. Find the height of the pole.

(a) 7.1 meter (b) 9 meter (c) 10 meter (d) 11 meter

16. When the elevation angle of the Sun increases from 45° to 60°, the length of a pole's shadow decreases by 3 meters. Determine the height of the pole (√3 = 1.732).

17. If the sun's elevation angle increases from 45° to 60°, the length of a pole's shadow decreases by x meters. Determine the height of the pole, assuming √3 = 1.732, rounded to three decimal places.

18. When the sun's angle of elevation changes from 45° to 60°, the length of the shadow of a telegraph pole changes by 4 meters. Determine the length of the shadow when the angle of elevation is 30°.

19. When the angle of elevation of the sun is 30°, the length of the shadow of a pillar is 9 meters. Find and write the height of the pillar.

20. When the angle of elevation of the sun increases from 45° to 60°, the length of a pole’s shadow shortens by 3 meters. Determine the height of the pole. [Take \(\sqrt{3} = 1.732\) and find the approximate value correct to three decimal places.]

21. A vertical pole of height 126 decimeters is bent at some point above the ground, causing its upper part to tilt and touch the ground, forming a 30° angle with the horizontal surface. Calculate and write how high from the ground the pole was bent and how far from the base the tip of the pole touches the ground.

22. When the angle of elevation of the Sun is ____________ than 45°, the length of a pillar’s shadow is less than the height of the pillar.

23. If the ratio of the length of a shadow of a pillar to the height of the pillar is √3:1, find the angle of elevation of the Sun.

24. A solid vertical cylindrical cone has both its diameter and height equal to 21 cm. Find the volume of the largest possible sphere that can be obtained from this cone. Also, determine the ratio of the volumes of the cone and the sphere.

25. The distance between two pillars is 150 meters. One is three times taller than the other. From the midpoint of the line segment connecting the bases of the two pillars, the angles of elevation to the tops of the pillars are complementary. Find the height of the shorter pillar.

26. A vertical circular cylinder has a height that is twice its radius. If the height were six times the radius, then the volume of the cylinder would be 539 cubic decimeters more. What is the height of the cylinder?

27. A 30-meter high pole and a pillar are located on the same horizontal plane. The foot of the pillar is seen from the top of the pole at a depression angle of 30°, and the top of the pillar is seen from the base of the pole at an elevation angle of 60°. What is the height of the pillar and the distance between the pole and the pillar?

28. A classroom needs to be built for 40 students in such a way that each student gets 5 square meters of floor area and 25 cubic meters of space. If the length of the room is 20 meters, what should be its height?

(a) 5 meters (b) 7 meters (c) 9 meters (d) 11 meters

29. From a point 50 meters away from the base of an incomplete pillar, the angle of elevation to its top is 30°. How much taller must the pillar be so that the angle of elevation to its new top from the same point becomes 45°?

30. A vertical hollow cylindrical pipe has an outer radius of 16 cm and an inner radius of 12 cm, with a height of 36 cm. If the pipe is melted and solid cylindrical rods are made from it, each with a diameter of 2 cm and a length of 6 cm, determine how many such solid rods can be produced.