Without using methods from calculus, find the optimal dimensions and , representing length and width, for a rectangle that must be contained within a unit semi-circle.
P.S: Rotate the rectangle by 90 degrees and re-examine the problem
Could be that my reasining is wrong but nevertheless I'l give it a try:
Since Archimedes it is known that a square is the optimal rectangle inscribed in a circle. The side length of this square is .
Now cut this figure parallel to one side of the square into 2 congruent parts: The circle becomes a semi-circle and the sqare is now a rectangle with the original side of the square as length and half of the length as height.
With your problem you have .
I've already completed this problem, I was putting it up as a challenge.
Assuming a rectanle that is inscribed with a semi cirlce has two of it's
corners intersecting (roughly speaking) the curve of the circle.
Since we want both width, , and height,
must be at an optimal point, allowing for the maximum area.
We can find:
The above is true because of the geometry of the problem.
If rotated 90 degrees h would be exactly one-half the total width.
and,
Resubbing, we get:
Above is true because oscillations in the sine function make
the maximum value of it to be 1.
Therefore,
...I think