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[cookiez69]

A piece of wire 7 metres long is cut into two parts, one is used to form a square, and the other to form a rectangle whose length is three times its width. Find the lengths of the two parts if the sum of the areas is a minimum.

Thanks for helping in advanced :]

 

[Sy123]

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|---------- x --------- ||-------(7-x)------|

A piece of wire is shown above cut into 2 pieces of lengths x and 7-x, since the wire is 7 metres long.

Now lets take x to be the wire cut into a square.

In such a square the perimeter must be x, hence since all sides are equal, each side length of our square is x/4

Hence the area of our square is (x/4)^2

Looking at the rectangle, first leave the piece of wire alone, and consider a rectangle where the length is three times the width.

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Let the dotted length be 3y, and the dashed width be y. (IGNORE the dots in the middle)

The perimeter of this will be: 3y + 3y + y + y = 7 - x

Hence it must follow that:



Now, we have constructed the shapes and now we need to find the areas of them:



Where A is the area

Lets sub in our x from before, it follows that



We need to find the minimum x





Now we equate this to zero and solve for x







Finding the second derivative to text our answers



Therefore the result we found is a minimum, hence the length of the two parts are: