Two ships are sailing on either sides of a lighthouse. The angle of elevation of the top of the lighthouse, as observed from the ships, are 60° and 45°, respectively. If the lighthouse is 200 m tall, find the distance between the two ships. (Assume √("3" ) = 1.732)
Two ships are sailing on either sides of a lighthouse. The angle of elevation of the top of the lighthouse, as observed from the ships, are 60° and 45°, respectively. If the lighthouse is 200 m tall, find the distance between the two ships. (Assume √("3" ) = 1.732)
Step 1: Assume the height of the lighthouse and the position of the two ships.
Let AB be the lighthouse and C and D be the positions of the two ships.
Step 2: Write the trigonometric ratio in ∆BAC.
Given AB = 200 m>.
Write the trigonometric ratio in ∆BAC.
tan 45° = 𝐴𝐵/𝐴𝐶
⇒ 1 = 200/𝐴𝐶
⇒ 𝐴𝐶 = 200 𝑚 >.
Write the trigonometric ratio in ∆BAD.
tan 60° = 𝐴𝐵/𝐴𝐷
⇒ √3 = 200/𝐴𝐷
⇒ 𝐴𝐷 = 200/√3
⇒ AD = 200/1.732 = 𝟏𝟏𝟓.𝟒𝟕 𝒎.
Step 3: Find the distance between the two ships.
𝐴𝐶 = 200 𝑚, AD = 115.47 𝑚
∴𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 between two ships = 𝐶𝐷
⇒ 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = 𝐴𝐶 + 𝐴𝐷
⇒ 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = 200 + 115.47
⇒ 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = 315.47 𝑚>
Hence, the distance between the two ships is 315.47 m.
Step 1: Assume the height of the lighthouse and the position of the two ships.
Let AB be the lighthouse and C and D be the positions of the two ships.
Step 2: Write the trigonometric ratio in ∆BAC.
Given AB = 200 m>.
Write the trigonometric ratio in ∆BAC.
tan 45° = 𝐴𝐵/𝐴𝐶
⇒ 1 = 200/𝐴𝐶
⇒ 𝐴𝐶 = 200 𝑚 >.
Write the trigonometric ratio in ∆BAD.
tan 60° = 𝐴𝐵/𝐴𝐷
⇒ √3 = 200/𝐴𝐷
⇒ 𝐴𝐷 = 200/√3
⇒ AD = 200/1.732 = 𝟏𝟏𝟓.𝟒𝟕 𝒎.
Step 3: Find the distance between the two ships.
𝐴𝐶 = 200 𝑚, AD = 115.47 𝑚
∴𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 between two ships = 𝐶𝐷
⇒ 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = 𝐴𝐶 + 𝐴𝐷
⇒ 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = 200 + 115.47
⇒ 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = 315.47 𝑚>
Hence, the distance between the two ships is 315.47 m.
