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We are dreaming building a house with a net (netto) area of $100m^2$.

Net means the thickness of the outer walls are not counted regarding the area.

Inner walls doesn't matter now.

The ratio needs to be $1 : 1.6$ regarding the outer sides.

The thickness of the outer wall is $40$ centimeters.

Example picture not proportional!: picture

The BIG question: what is the function for getting all the possible outer side lengths? The ones marked with "?" in the example picture.

From this, many would benefit, since "Olgyay", a bigger architect said passive houses in the temperate zone are the most energy-saving if their side ratio is 1:1.6

Siong Thye Goh
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Peter
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1 Answers1

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Let's say the shorter side of the house has length $x$ (on the outside). This means the longer side has length $1.6\cdot x$.

The inner rectangle, therefore, has sides of length $x-0.8$ and $1.6x - 0.8$, so the equation you want to solve is

$$(x-0.8)\cdot (1.6\cdot x - 0.8) = 100$$

This is a regular quadratic equation that can easily be solved using the standard technicque:

  • Multiply out the left
  • Bring everything to the left side
  • Change the equation to the form $ax^2+bx+c=0$
  • Remember that if $ax^2+bx+c=0$, then $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
5xum
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  • wow, thanks! but where is the 40 cm thick outside wall in this? – Peter Aug 17 '17 at 07:59
  • @Peter The inner rectangle has side $x-0.8$ because $0.4$ is taken up by the wall on each side. – 5xum Aug 17 '17 at 08:00
  • I understand now, thanks :) – Peter Aug 17 '17 at 08:02
  • @Peter You know, if the answer is what you were looking for, you should also accept it (by clicking on the little grey tick icon on the left side of the question) – 5xum Aug 17 '17 at 08:06
  • If I input it to Wolfram Alpha: https://www.wolframalpha.com/input/?i=(x%E2%88%920.8)%E2%8B%85(1.6%E2%8B%85x%E2%88%920.8)%3D100

    It says 7.2m * 8.5m. Those aren't in 1:1.6 ratio.

    Your answer is probably correct, only I am the one who doesn't understand the whole part :)

     – Peter Aug 17 '17 at 08:10
  • @Peter No, it doesn't say 7.28.5. It says that the possible values of $x$* are either $-7.25712$ or $8.55712$. Now given that $x$ is a positive number, you know which of those is correct. And then you know $x$, and you use the second sentence of my answer to calculate the other side. – 5xum Aug 17 '17 at 08:13