The printed matter on a 12 by 24 centimeter page of a book must cover 64 square centimeters. If all margins are to be the same width, how wide should the margins be?
The printed matter on a 12 by 24 centimeter page of a book must cover 64 square centimeters. If all margins are to be the same width, how wide should the margins be?
So we need a few formulas to start.
The area of a rectangle is $\displaystyle A=L \cdot W$
We know the area $\displaystyle A=84 in^2$
We also know that the length is five more than the width so
$\displaystyle L=W+5$ and that the width is W.
Now we evaluate the area formula to get
$\displaystyle 84=(W+5)W \iff 84=W^2+5W \iff 0=W^2+5W-84 $
$\displaystyle 0= (W+12)(W-7)$
So now using the zero product priciple we set each factor equal to zero to get
$\displaystyle W+12=0 \iff W=-12$ and $\displaystyle W-7=0 \iff W=7$
Since we are talking about the width of a frame the negative solution doesn't make sense so we are left with the width of the frame is 7 in.
Good luck.
Let x be the size of the margins. Then the length of the printed matter is $\displaystyle 24-2x$ and the width is $\displaystyle 12-2x$
Using the Area formula from the above post we get
$\displaystyle 64=(24-2x)(12-2x)$ we can factor a 2 out of each factor on the left hand side to get
$\displaystyle 64=2\cdot 2\cdot (12-x)(6-x)$ now we can reduce the equation by dividing both sides by 4 to get
$\displaystyle 16=(12-x)(6-x)$ Now we expand the right hand side to get
$\displaystyle 16=72-18x+x^2 \iff x^2-18x+56=0 \iff (x-14)(x-4)=0$
So we get that x=14 or x=4
We reject the first solution becuase if we plug it back into the original equation the length and width are negative so x =4 is the solution.
The margins should be 4cm