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the curve with equation y=x(6-x) crosses the x-axis at the origin O and at the point R. the rectangle OPQR is such that PQ is tangent to the curve at its maximum point.
1. show that the area of the region between the curve and the x-axis is two thirds of the area of the rectangle OPQR.
a straight line through the origin meets the curve at the point A with x-coordinate a.
2. find, in terms of a, the area of the shaded region between the curve and the line. hence find, correct to three significant figures, the value of a for which this area is half the area of the rectangle OPQR.
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