a page of print is to contain24in^2 of printed region,a margin of 1and1/2 at the bottom,and a margin of 1in.at the sides..what are the dimensions of the smallest page that will fill these requirements?

Question

anonymous · Answer

http://tutorial.math.lamar.edu/Classes/CalcI/Optimization.aspx See example 6

radar · Answer

A great reference.  Thanks godnod.  Let me see if I can put it together and held ronadelossantos.

Let y equal one dimension of the printed material.  Let x = the other dimension of the printed material.  It is stated that the printed material is 24 sq. inches.  Thus: xy=24.  We will use that information later.

One dimension of the paper (including the margins) will be x+1 and the other dimension is y+2.   Returning to the xy=24 we can say y=24/x.  We now have enough information to solve.

radar · Answer

$$A=(x+1)((x/24)+2)$$ This becomes:
$$24x/x+2x+24/x+2$$
$$24+2x+24/x+2$$
$$2x+24/x+26=A$$
$$2x+24x ^{-1}+26=Area$$
Differntiating and set zero for min/max

radar · Answer

A' = 0$$A'=2-24x ^{-2}=0$$

radar · Answer

$$-24=-2x ^{2}$$
$$x ^{2}=12$$
$$x=2\sqrt{3}$$ one dimension solve for y

y=24/x
$$y=24/(2\sqrt{3}=12/\sqrt{3}=4\sqrt{3}$$