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Given just this diagram, I am trying to find the dimensions which gives the maximum area. enter image description here

I understand that I have to apply the first derivative test for local extrema, which involves setting the first derivative of a function equal to $0$ to find the critical points of the function which could be a local/absolute maxima/minima. We then use the second derivative test to confirm whether the second derivative is less than 0 in the domain of the function and if it aligns, means it is the maximum dimensions.

The perimeter of the rectangle is: $P = 2h + 2w$

with no other information given, how do I express the an expression as a function of 1 variable so that I can perform the first and second derivative test?

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1 Answer 1

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Hint:

The radius of the semicircle is always constant, including at where the vertices of the rectangle meet the semicircle.

From there, you can solve for either $h$ or $w$ and then take the positive square root as $h, w > 0$.

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  • $\begingroup$ What do you mean by that? I cant seem to put it together into a mathematical equation $\endgroup$
    – user307640
    Oct 9, 2022 at 7:14
  • $\begingroup$ $h^2 + (w/2)^2$ is... $\endgroup$
    – Toby Mak
    Oct 9, 2022 at 12:19

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