Question 1. Show Question 2. Solution: AB = 20 m, BC = \(20 \sqrt{3}\) m, θ = ? In ∆ABC, Question 3. Solution: BD = AB – AD = 6 – 2.54 = 3.46 m In rt., ∆DBC, sin 60° = \(\frac{B D}{D C}\) \(\frac{\sqrt{3}}{2}=\frac{3.46}{\mathrm{DC}}\) \(\sqrt{3}\)DC = 3.46 x 2 ∴ Length of the ladder, DC = \(\frac{6.92}{\sqrt{3}}=\frac{6.92}{1.73}\) DC = 4 m Question 4. ∴ Length of ladder, AC = 5 m 2.5 m Question
5. tan θ = \(\frac{30}{10 \sqrt{3}}\) tan θ = \(\sqrt{3}\) ⇒ tan θ = tan 60° ∴ θ = 60° Question 6. Some Applications of Trigonometry Class 10 Important Questions Short Answer-I (2 Marks)Question 7. ∴ Height of the light house = 273 m Question 8. ∴ Height of the first pole, AB = CD = CE – DE = 24 – 8.66 = 15.34 m Question 9. ∴ Distance between the two ships, CD = BD – BC = 103.92 – 60 = 43.92 m Question 10. Let AB be the light house and C and D be the two ships, In rt. ∆ABC, tan 45° = \(\frac{\mathrm{AB}}{\mathrm{BC}}\) Question 11. Question 12. ∴ Height of the tower, DE = BC DE = AC – AB DE = 60 – 20\(\sqrt{3}\) = 20(3 – \(\sqrt{3}\)) DE = 20(3 – 1.73) = 20(1.27) DE = 25.4 m Question 13. Let height of building be x and the distance between tower and building be y. In ∆ABC, ∴Height of the building is 10\(\sqrt{3}\) m. Question 14. Question 15. Let AE be the building and CD be the tower. Let height of the tower = h m and, the horizontal distance between tower and building = x m …[Given BD = AE = 50 m ∴ BC = CD – BD = (h – 50) m From (i), x = h – 50 = 118.25 – 50 = 68.25 m Height of the tower, h = 118.25 m ∴ Horizontal distance between tower and Building, x = 68.25 m Question 16. Let the man standing on the deck of a ship be at point A and let CE be the hill. Here BC is the distance of hill from ship and CE be the height of hill. In rt. ∠ABC, tan 30° = \(\frac{\mathrm{AB}}{\mathrm{BC}}\) BC = 10\(\sqrt{3}\) m .(i) BC = 10(1.73) = 17.3 m …[:: \(\sqrt{3}\) = 1.73 AD = BC = 10 \(\sqrt{3}\) m …(ii) [From (i) In rt. ∆ADE, tan 60° = \(\frac{\mathrm{DE}}{\mathrm{AD}}\) ⇒ \(\sqrt{3}=\frac{\mathrm{DE}}{10 \sqrt{3}}\) … [From (ii) ⇒ DE = 10\(\sqrt{3}\) × \(\sqrt{3}\) = 30 m ∴ CE = CD + DE = 10 + 30 = 40 m Hence, the distance of the hill from the ship is 17.3 m and the height of the hill is 40 m. Some Applications of Trigonometry Class 10 Important Questions Long Answer (4 Marks)Question 17. \(\sqrt{3}\).\(\sqrt{3}\) (y – 10) = y … [From (i) 3(y – 10) = y ⇒ 3y – 30 = y 3y – y = 30 ⇒ 2y = 30 ∴ Height of the tower, y = 15 m Question 18. ∴ Distance between the cars, CD = BD + BC = 173 + 100 = 273 m Question 19. ⇒ 3y = 100\(\sqrt{3}\) – y ⇒ 4y = 100\(\sqrt{3}\) ∴ Height of the poles, y = \(\frac{100 \sqrt{3}}{4}\) = 25\(\sqrt{3}\) m = 25(1.73) = 43.25 m Question 20. ∴ Height of transmission tower = 20 m Question 21. ∴ Height of the tower, h = 236.5 m Question 22. tan 30° = \(\frac{\mathrm{AB}}{\mathrm{BE}}\) \(\frac{1}{\sqrt{3}}=\frac{y-12}{x}\) x = \(\sqrt{3}\)(y – 12) In rt. ∆ACD, tan 60° = \(\frac{\mathrm{AC}}{\mathrm{DC}}\) ⇒ \(\frac{\sqrt{3}}{1}=\frac{y}{x}\) ⇒ \(\sqrt{3}\)x = y ⇒ \(\sqrt{3}\).\(\sqrt{3}\) (y – 12) = y … [From (i) ⇒ 3 (y – 12) = y ⇒ 3y – 36 = y ⇒ 3y – y = 36 ⇒ 2y = 36 ⇒ y = 18 ∴ Height of the multi-storeyed building, y = 18 m Question 23. Question 24. \(\frac{1}{\sqrt{3}}=\frac{50}{x}\) x = 50\(\sqrt{3}\) In rt. ∆BCD, tan 60° = \(\frac{\mathrm{CD}}{\mathrm{BC}}\) \(\sqrt{3}\) = \(\frac{\mathrm{H}}{50 \sqrt{3}}\) ⇒ H = 50\(\sqrt{3}\) × \(\sqrt{3}\) = 150 ∴ Height of the hill = 150 m Question 25. tan 45° = \(\frac{A B}{B C}\) = 1 = \(\frac{h}{x}\) = h = x … (1) In rt. ∆ABD, tan 30° = \(\frac{AB}{BD}\) \(\frac{1}{\sqrt{3}}=\frac{h}{x+1}\) \(\sqrt{3}\)h = x + 1 \(\sqrt{3}\)x = x + 1 …[From (i) \(\sqrt{3}\)x – x = 1 (\(\sqrt{3}\) – 1)x = 1 Hence, the height of the hill is 1.365 km. Question 26. ∴ Height of the tower = 47.3 m Question 27. Question 28. Question 29. ∴ (i) Difference between the heights = 20 m and (ii) Distance = 34.64 m Question 30. ∴Height of the building, CD = 20 m Question 31. Length of the flagstaff, RS = 17.3 – 10 = 7.3 m Question
32. Height of pole = 34.6 m Puting the value of y in (i), x = \(\sqrt{3}\) (20\(\sqrt{3}\)) ∴ x = 60 ∴ CD, x = 60 m Distance of poin from the pole and BC = 80 – x = 80 – 60 = 20 m Question 33. Question
34. ∴ Difference between the heights of the building and the tower, DE = 20 m Distance between them, BC= 20\(\sqrt{3}\)m = 20(1.73) = 34.6 m Question 35. Lęt PH be the pillar. Let the distance from the hole to the place where snake is caught = x m Let P be the top of the pillar and S be the point where the snake is ∴ SC = (27 – x)m SC = PC = (27 – x)m …[∵ Their speeds are equal In rt. ∆PHC PH2 + CH2 = PC2 …[Pythagoras’ theorem 92 + x2 = (27 – x)2 81 + x2 = 729 – 54x + x2 54x = 729 – 81 = 648 x = \(\frac{648}{54}\) = 12 m Hence, required distance, x =12 m Question 36. 1 = \(\frac{BC}{120}\) BC = 120 m …(i) In rt. ∆ABD, tan 60° = \(\frac{BD}{AB}\) \(\sqrt{3}\) = \(\frac{BD}{120}\) BD = 120 \(\sqrt{3}\) …(ii) Height of the flagstaff, CD = BD – BC = 120\(\sqrt{3}\) – 120 = 120(\(\sqrt{3}\) – 1) = 120(1.73 – 1) = 120(0.73) = 87.6 m Question 37. (i) \(\frac{x}{y}\) = tan 30° = \(\frac{1}{\sqrt{3}}\) y = \(\sqrt{3}\)x ..(i) In ∆ABC, (ii) \(\frac{x+5}{y}\) = tan 60° tan 60° = \(\sqrt{3}\) \(\frac{x+5}{\sqrt{3} x}\) = \(\sqrt{3}\) …[From (i) ⇒ 3x = x + 5 or x = 2.5 ∴ Height of Tower = x = 2.5 m Question 38. Putting the value of h in equation (i), ∴ x = 20\(\sqrt{3}\) m Using pythagoras’ theorem, AC = \(\sqrt{(h)^{2}+(x)^{2}}\) ∴ AC = \(\sqrt{(20)^{2}+(20 \sqrt{3})^{2}}\) = 40 m Distance of the cloud from A = 40 m. Question 39. ⇒ 3H – 180 = H + 60 ⇒ 3H – H = 60 = 180 ⇒ 2H = 240 ⇒ H = 120 ∴ Height of the cloud = 120 m Question 40. Let BC be the tree In rt. ∆ABC, tan 45° = \(\frac{BC}{AB}\) Question 41. Let the height of the tower, AB = h m In ∆ABC, tan 60° = \(\frac{AB}{AC}\) \(\sqrt{3}\) = \(\frac{h}{4}\) ⇒ h = 4\(\sqrt{3}\) Hence, height of the tower = 473 m Question 42. Hence, height of the tower PQ is 96.54 m and the distance PX is 109.3 m. Question 43. ∴ Distance travelled by ship is 115.46 m Question
44. Let CD be height of an aeroplane flying above the ground and AB be the two banks of the river. In rt. ∆BCD, tan 60° = \(\frac{CD}{BC}\) The width of the river, AB = AC – BC = 300 – 173.2 = 126.8 m Question 45. Let BD be the tower, i.e., x m. In rt. ∆ABD, tan 60° = \(\frac{BD}{AB}\) From (i) x = \(\sqrt{3}\) (20\(\sqrt{3}\)) = 60 m ∴Height of the tower, r = 60 m Required horizontal distance, y = 20\(\sqrt{3}\)m y = 20(1.73) = 34.6 m …[∵ \(\sqrt{3}\) = 1.73 Important Questions for Class 10 Maths |