A windows of fixed perimeter (including ...

A windows of fixed perimeter (including the base of the arc) is in the form of a rectangle surmounted by a semi- circle. The semi- circular portion is fitted with coloured glass while the rectangular portion is fitted with clear glass.
The clear glass transmits three times as much light per square metre as the coloured glass does. what is the ratio of the sides of the rectangle so that the window transmits the maximum light?

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A window of perimeter P (including the base of the arch) is in the form of a rectangle surrounded by a semi-circle. The semi-circular portion is fitted with the colored glass while the rectangular part is fitted with the clear glass that transmits three times as much light per square meter as the colored glass does. What is the ratio for the sides of the rectangle so that the window transmits the maximum light?

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