Related rates, Calc. I problem

[tangokilla]

The Almajest company is seeking to track aircraft that flies through it property. Th company has two radar stations, one in Alzeda, and one in Busy. Busby is one mile due east of Alzeda. Ffr an aircraft each station can detect the aircraft's distance from the station and the rate at which this distance is changing. So for any aircraft, four measurements are taken. The company wants to be able to determine the location of a plane measured according to a specific grid. Since the main offices of Almajest Company are in Alzeda, the origin is to be at the Alzeda station, one axis is to run due east (and west) of Alzeda, and the other axis is to run due north (and south) of Alzeda. In addition, the company wants to determine an aircraft's velocity broken into its east-west and its north-south components.

Assume that altitude of the aircraft can be ignored, determine the four formulas required for the company.

Any help would be appreciated. Thanks!

[HallsofIvy]

Okay, set up a coordinate system so that Alzeda is at (0,0), Busby is at (1, 0). The airplane will be at point (x,y). Let the distance from Alzeda to the airplane be d1 and the distance from Busby to the airplane be d2. Then we have x^2+ y^2= d1^2 and (x-1)^2+ y^2= d2^2. Subtracting the second equation from the first, we have x^2- (x-1)^2= d1^2- d2^2. Multiplying (x-1)^2, we get 2x+ 1= d1^2- d2^2 so x= (d1^2- d2^2- 1)/2. Put that back into x^2+ y^2= d1^2 to calculate y. (y can be positive or negative. Just knowing the distance from each place to the airplane does not tell you whether is is north or south of them.)

Again, x^2+ y^2= d1^2 so 2x x'+ 2y y'= 2d1 d1'. Also (x-1)^2+ y^2= d2^2 so 2(x-1)x'+ 2yy'= 2d2 d2' We have x x'+ y y'= d1 d1' and (x-1)x'+ yy'= d2 d2'. I'll let you solve those two equations for x' and y'. (d1, d1', d2, and d2' are given by the equipment. x and y are calculated as above.)