Hello, for qn5: how can one tell that the area of the segment is 1/3 the area of the circle?
Well actually the segment I have drawn is the one that contains the vertical line hatching, and this is 2/3 the area of the circle. But the other segment is 1/3 the area.
You can tell this because we can see that the angle I have labelled in the diagram is π/3 (because it is arcsin(0.5R / R)). By symmetry its mirror image in the x-axis is also π/3, and therefore the sum of the two is 2π/3; since the total ange is 2π this means the segment area is ((2π/3)/2π) tims the area of the circle, which is 1/3 the area of the circle.
What was your thought process when choosing the equations for the sides of the rectangles?
Hi Kara, thanks for your question.
It’s a good question, and actually there is nothing special about the method I have chosen. I could just as well have said that one side is x, and then the adjacent side must be L/2-x. Then when you work out the area and differentiate it you get x = L/4 at the maximum, and its adjacent side is L/4. So if anything this would be neater…
I think the necessary thought process for answering this question in a reasonably rigorous way is ‘Find any equation for the area in terms of some unknown and L; solve for the unknown at the maximum to find the value of the unknown in terms of L; plug in the solved value to find the area in terms of L’. But there are absolutely no rules about how to express the line lengths to generate the initial equation: anything that works is good.
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