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 tmac20 November 1st, 2011 01:26 PM

Finding the width of a uniform border

A mural is to be painted on a wall that is 15 units long and 12 units high. A border of uniform width is to surround the mural. If the mural is to cover 75% of the area of the wall, how wide must the border be?

 The Chaz November 1st, 2011 01:32 PM

Re: Finding the width of a uniform border

The "75%..." part is the easiest to tackle.

The wall has area 12 x 15 = 180 (square units), so we want our mural to have area .75*180 = 135 (square units).

Now we determine the dimensions of the mural.

The requirement of a "uniform border" means that the mural will have dimensions (15 - 2x)*(12 - 2x), where x is the width of the border.
^ deducing this is often the hardest part of problems like this, so draw a picture and see why my claim is true.

So (15 - 2x)*(12 - 2x) = 135 is the equation for the area of the mural.

FOIL, get in standard form, then factor/quadratic formula.

 MarkFL November 1st, 2011 01:33 PM

Re: Finding the width of a uniform border

Let W be the width and H be the height of the wall, and B be the width of the border. We then require:

$$$W-2B$$$$H-2B$$=\frac{3WH}{4}$

$WH-2BW-2BH+4B^2=\frac{3WH}{4}$

$4B^2-2(W+H)B+\frac{WH}{4}=0$

$16B^2-8(W+H)B+WH=0$

Application of the quadratic formula yields:

$B=\frac{8(W+H)\pm\sqrt{64(W+H)^2-64WH}}{32}=\frac{W+H\pm\sqrt{W^2+H^2+WH}}{4}$

Letting W = 15 m and H = 12 m, we find:

$B\approx0.89,12.61$

The only root that makes sense is $B\approx0.89$

 CuhCalvin November 1st, 2011 03:57 PM

Re: Finding the width of a uniform border

ZOMG you don't know? so nooby

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