Question 13 - Case Study - CBSE Class 10 Sample Paper for 2022 Boards - Maths Basic [Term 2] - Solutions of Sample Papers for Class 10 Boards

Last updated at April 16, 2024 by Teachoo

A lighthouse is a tall tower with light near the top. These are often built on islands coasts or on cliffs. Lighthouses on water surface act as a navigational aid to the mariners and send warning to boats and ships for dangers. Initially wood, coal would be used as illuminators. Gradually it was replaced by candles, lanterns, electric lights. Nowadays they are run by machines and remote monitoring.Prongs Reef lighthouse of Mumbai was constructed in 1874-75. It is approximately 40 meters high and its beam can be seen at a distance of 30 kilometers. A ship and a boat are coming towards the lighthouse from opposite directions. Angles of depression of flash light from the lighthouse to the boat and the ship are 30° and 60°respectively.

i) Which of the two, boat or the ship is nearer to the light house. Find its distance from the lighthouse?

ii) Find the time taken by the boat to reach the light house if it is moving at the rate of 20 km per hour.

Transcript

A lighthouse is a tall tower with light near the top. These are often built on islands coasts or on cliffs. Lighthouses on water surface act as a navigational aid to the mariners and send warning to boats and ships for dangers. Initially wood, coal would be used as illuminators. Gradually it was replaced by candles, lanterns, electric lights. Nowadays they are run by machines and remote monitoring. Prongs Reef lighthouse of Mumbai was constructed in 1874-75. It is approximately 40 meters high and its beam can be seen at a distance of 30 kilometers. A ship and a boat are coming towards the lighthouse from opposite directions. Angles of depression of flash light from the lighthouse to the boat and the ship are 30° and 60° respectively.
i) Which of the two, boat or the ship is nearer to the light house. Find its distance from the lighthouse?
ii) Find the time taken by the boat to reach the light house if it is moving at the rate of 20 km per hour.
Since Angle of depression = Angle of elevation
∴ ∠ ADB = 30° and ∠ ACB = 60°
We need to find BD and BC
In right triangle ΔABD,
tan D = (𝑆𝑖𝑑𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑡𝑜 𝑎𝑛𝑔𝑙𝑒" " 𝐷)/(𝑆𝑖𝑑𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 𝑎𝑛𝑔𝑙𝑒" " 𝐷)
tan 30° = 𝐴𝐵/𝐵𝐷
(" " 1)/√3 = (" " 40)/𝐵𝐷
BD = 40√𝟑 m
In right triangle ΔABC,
tan C = (𝑆𝑖𝑑𝑒 𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒 𝑡𝑜 𝑎𝑛𝑔𝑙𝑒" " 𝐷)/(𝑆𝑖𝑑𝑒 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 𝑎𝑛𝑔𝑙𝑒" " 𝐷)
tan 60° = 𝐴𝐵/𝐵𝐶
√3 = (" " 40)/𝐵𝑐
BC = 𝟒𝟎/√3 m
Thus, ship at point C is nearer
And, it’s distance from lighthouse is
BC = 40/√3 m
= 40/√3 ×√3/√3 m
= (𝟒𝟎√𝟑)/𝟑 m
ii) Find the time taken by the boat to reach the light house if it is moving at the rate of 20 km per hour.
Boat is at point D
We need to find time taken by the boat to reach the light house if it is moving at the rate of 20 km per hour.
Since speed as at km/hr and distance is in meters
We convert Distance into km
Distance = BD = 40√3 m
= (𝟒𝟎√𝟑)/𝟏𝟎𝟎𝟎 km
Now,
Time Taken = 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒/𝑆𝑝𝑒𝑒𝑑
= 𝑩𝑫/𝟐𝟎
= ((40√3)/1000)/20
=(40√3)/(1000 × 20)
= (𝟐√𝟑)/𝟏𝟎𝟎𝟎 hours
= (2√3)/1000 × 60 minutes
= (𝟏𝟐√𝟑)/𝟏𝟎𝟎 minutes
Since speed as at km/hr and distance is in meters
We convert Distance into km
Distance = BD = 40√3 m
= (𝟒𝟎√𝟑)/𝟏𝟎𝟎𝟎 km
Now,
Time Taken = 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒/𝑆𝑝𝑒𝑒𝑑
= 𝑩𝑫/𝟐𝟎
= ((40√3)/1000)/20
= (40√3)/(1000 × 20)
= (𝟐√𝟑)/𝟏𝟎𝟎𝟎 hours
= (2√3)/1000 × 60 minutes
= (𝟏𝟐√𝟑)/𝟏𝟎𝟎 minutes

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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