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A large window consists of a rectangle with an equilateral triangle resting on its top. If the perimeter of the window is 33 feet, find the dimensions of the rectangle that will maximize the area of the window.

Charity
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Denote the upper side of the rectangle with $x$ and the other side with $y$. Then you are given that the perimeter of the whole window (rectangular area and triangle) is equal to $33$, i.e. $$\underbrace{2x+2y}_{\text{rectangle}}+\underbrace{3x}_{\text{triangle}}-\underbrace{2x}_{\text{inner side common to both}}=33 \iff 3x+2y=33$$ or equivalently that $$y=\frac{33}{2}-\frac{3}{2}x$$ The area of the equilateral triangle is given by $$\frac{\sqrt{3}}{4}x^2$$ and the area of the rectangle is given by $$x\cdot\left(\frac{33}{2}-\frac{3}{2}x\right)$$ Thus the total are that you need to maximize by the choice of $x$ is equal to $$\frac{\sqrt{3}}{4}x^2+x\cdot\left(\frac{33}{2}-\frac{3}{2}x\right)=\left(\frac{\sqrt{3}-6}{4}\right)x^2+\frac{33}{2}x$$ This can be done with the usual method of the first and second derivative. Can you take it from here?

Jimmy R.
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  • does 2x+2y represent the dimensions of the rectangular portion of the window? If so, I do not think we can say it equals 33, since the dimensions of the entire window - the rectangular and triangular portion - is 33 feet. Perhaps I am misreading something. – 123 Dec 16 '14 at 14:46
  • @mathtastic Oh, you are correct. I misread it, not you. I will edit my answer – Jimmy R. Dec 16 '14 at 14:48
  • But don't the equilateral triangle and rectangle share a side, which does not contribute to the perimeter of the window? I get a perimeter of 3x + 2y. See http://imgur.com/tEd05hy. – Tebbe Dec 16 '14 at 14:56
  • If the shape is closed then would the perimeter be 2y+3x where x is the length and y the width (that is, y the shorter length.)? This would give us the one exposed length of the rectangle, the rectangle's two short sides and then the two sides of the triangle connecting to the rectangle. – 123 Dec 16 '14 at 14:57
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    @mathtastic Oh, if there is a possibility to make a mistake, I make it. I will better do something else, many thanks for the corrections. – Jimmy R. Dec 16 '14 at 15:00
  • @Tebbe Thanks for the correction and the nice shape in the link. – Jimmy R. Dec 16 '14 at 15:01
  • Hello, can someone finish the problem? – Charity Jan 06 '15 at 01:55