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On a dark night, two ships, Saga and Hero, sail parallel to a straight coastline on which there are two lights of equal brightness, 16 kilometres apart. P(x, b) b (-8, 0) (8, 0) x

On a dark night, two ships, Saga and Hero, sail parallel to a straight coastline on which there are two lights of equal brightness, 16 kilometres apart. P(x, b) b (-8, 0) (8, 0) x

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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On a dark night, two ships, Saga and Hero, sail parallel to a straight coastline on which there are two lights of equal brightness, 16 kilometres apart. P(x, b) (-8, 0) (8, 0) x Suppose the coastline is represented by the x axis where the origin O is chosen to be the midpoint of the light sources. It is known that the (total) brightness from the lights on a ship at point P(x, b) is 1 I = b² +(x+8)² 'b² +(x-8)² + dI (i) Show that dx 2P where P= (x+8)(3² +(x – 8)*)* +(x -8)(s² +(x + 8)*)* | and Q = (3° +(x +8)*)} (s? +(x-87)'. To answer parts (ii) and (iii), you may assume the following factorisation, given by a computer package, that P= 2x(x² + 64 + b? +16/64 + b² (x² + 64 + b² –16/64 + b²). (ii) Saga sails parallel to the coast at a distance 15 km from the coast. dI show that, as Saga sails from left to right, the brightness By considering on Saga increases to a maximum when x= 0 and then decreases. dx (iii) Hero sails parallel to the coast at a distance 6 km from the coast. Describe how the brightness on Hero changes as Hero sails from left to right. Give clear reasons for your answer.
Transcribed Image Text:On a dark night, two ships, Saga and Hero, sail parallel to a straight coastline on which there are two lights of equal brightness, 16 kilometres apart. P(x, b) (-8, 0) (8, 0) x Suppose the coastline is represented by the x axis where the origin O is chosen to be the midpoint of the light sources. It is known that the (total) brightness from the lights on a ship at point P(x, b) is 1 I = b² +(x+8)² 'b² +(x-8)² + dI (i) Show that dx 2P where P= (x+8)(3² +(x – 8)*)* +(x -8)(s² +(x + 8)*)* | and Q = (3° +(x +8)*)} (s? +(x-87)'. To answer parts (ii) and (iii), you may assume the following factorisation, given by a computer package, that P= 2x(x² + 64 + b? +16/64 + b² (x² + 64 + b² –16/64 + b²). (ii) Saga sails parallel to the coast at a distance 15 km from the coast. dI show that, as Saga sails from left to right, the brightness By considering on Saga increases to a maximum when x= 0 and then decreases. dx (iii) Hero sails parallel to the coast at a distance 6 km from the coast. Describe how the brightness on Hero changes as Hero sails from left to right. Give clear reasons for your answer.
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