Differential Equation luminescence qh/(h^2+(x−x0)^2+(y−y0)^2)^3/2

[tomtom1029]

hello, i'm really stumped on the final question, some direction or help would be much appreciated of where to even begin. attached an image and the question worded.

A sports stadium is lit by four floodlights standing at the four corners of a rectangle which contains the rectangular pitch placed symmetrically inside it. The length of the rectangle is 148 metres and the width is 56 metres. This question is concerned with finding the common optimal height for the floodlights giving 'best' illumination of the pitch. A coordinate system is set up with the origin at the centre of the pitch. The [FONT=MathJax_Math-italic]x[/FONT] axis points along the pitch and the [FONT=MathJax_Math-italic]y[/FONT] axis points across the pitch. The luminance [FONT=MathJax_Math-italic]I[/FONT] produced at a point [FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math-italic]y[/FONT][FONT=MathJax_Main])[/FONT]by a single light of power [FONT=MathJax_Math-italic]q[/FONT] positioned at a height [FONT=MathJax_Math-italic]h[/FONT] above the point [FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math-italic]y[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main])[/FONT] on the pitch is given by [FONT=MathJax_Math-italic]q[/FONT][FONT=MathJax_Math-italic]h/[/FONT][FONT=MathJax_Size2]([/FONT][FONT=MathJax_Math-italic]h^[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main])^[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]y[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Math-italic]y[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main])^[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Size2])[/FONT][FONT=MathJax_Main]3/[/FONT][FONT=MathJax_Main]2[/FONT] The luminance at any point on the pitch is given by the sum of the luminances at that point from each light. The power for each light is 1,202,650 units

(a)	Find the value of [FONT=MathJax_Math-italic]h

for which the luminance at the centre of the pitch takes its maximum value. Type in the number only and give your answer to AT LEAST THREE DECIMAL PLACEs. (8 marks)

[/FONT]

A sports stadium is lit by four floodlights standing at the four corners of a rectangle which contains the rectangular pitch placed symmetrically inside it. The length of the rectangle is 148 metres and the width is 56 metres. This question is concerned with finding the common optimal height for the floodlights giving 'best' illumination of the pitch. A coordinate system is set up with the origin at the centre of the pitch. The [FONT=MathJax_Math-italic]x[/FONT] axis points along the pitch and the [FONT=MathJax_Math-italic]y[/FONT] axis points across the pitch. The luminance [FONT=MathJax_Math-italic]I[/FONT] produced at a point [FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math-italic]y[/FONT][FONT=MathJax_Main])[/FONT]by a single light of power [FONT=MathJax_Math-italic]q[/FONT] positioned at a height [FONT=MathJax_Math-italic]h[/FONT] above the point [FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main],[/FONT][FONT=MathJax_Math-italic]y[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main])[/FONT] on the pitch is given by [FONT=MathJax_Math-italic]q[/FONT][FONT=MathJax_Math-italic]h[/FONT][FONT=MathJax_Size2]([/FONT][FONT=MathJax_Math-italic]h[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Math-italic]x[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Main]+[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]y[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Math-italic]y[/FONT][FONT=MathJax_Main]0[/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]2[/FONT][FONT=MathJax_Size2])[/FONT][FONT=MathJax_Main]3[/FONT][FONT=MathJax_Main]2[/FONT] The luminance at any point on the pitch is given by the sum of the luminances at that point from each light. The power for each light is 1,202,650 units

(a)	Find the value of [FONT=MathJax_Math-italic]h

for which the luminance at the centre of the pitch takes its maximum value. Type in the number only and give your answer to AT LEAST THREE DECIMAL PLACEs. (8 marks)

[/FONT]