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So, this is a Question 10 of the 2 unit (just 2 unit) mathematics HSC past paper (2002).
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. Suppose that the coastline is represented by the x axis where the origin ) 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...
I = 1 / (b^2 + (8 + x)^2) + 1 / (b^2 + (x - 8)^2)
(i) Show that dI / dx = -2P / Q Where...
P = [(x + 8)(b^2 + (x - 8)^2)^2 + (x - 8)(b^2 + (x - 8)^2)^2]
Q = (b^2 + (x + 8)^2)^2(b^2 + (x - 8)^2)^2
To answer part (ii) and (iii), you may assume the following factoirsation, givn by a computer package, that
P = 2x(x^2 + 64 + b^2 + 16√(64 + b^2))(x^2 + 64 + b^2 - 16√(64 + b^2))
(ii) Saga sails parallel to the coast at a distance 15 km from the coast. By considering dI / dx, show that, as Saga sails from left to right, the brightness on saga increases to a maximum when x = 0 then decreses.
(iii) Hero sails parallel to the coast at a distance 6 km from the coast. Describe how the brightness of hero changes as hero sails from left to right. Give reasons for your answer.
This is actually one of the easier questions!
1 x 8 - 3 + 0 = 5
Topic: Counting to 1,000,000 (Read 3188737 times)