## Partial Differentiation to Optimisation

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 162 metres and the width is 112 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 axis points along the pitch and the axis points across the pitch.

The luminance produced at a point by a single light of power positioned at a height above the point on the pitch is given by

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 1,548,200 units.

i need to find out the value of for which the luminance at the centre of the pitch takes its maximum value.
I also need,after finding h, to find the x coordinate of the darkest point on the pitch. and the y coordinates too.
i'm able to find h though.it turned out to be 69.6 but im not able to workout the required x and y coordinates of the darkest point on the pitch.
can anyone help?
cheers.