Question

A rectangular page is to contain 165 square inches of print. The top and bottom margins are each 0.5 inches wide , and the margins on each side is 1.75 inches wide. What should the dimensions be if the least amount of material is to be used

Answer 1

how did you get area=169?

Answer 2

@d92292 oops I misread. Hmm... let's try again. Let \(x\) be our width, \(y\) our height for the print area of the page. So we know \(xy=165\) so that we have \(165\,\text{in}^2\) for our print.

Now we're interested in minimizing the *total* area of the page, so consider \(A=(x+3.5)(y+1)\), which should make sense since we have two top/bottom margins of \(0.5\,\text{in}\) as well as two left\right margins of \(1.75\,\text{in}\). We're interesting in minimizing this function.$$A=xy+x+3.5y$$Recalling that \(xy=165\) and therefore \(y=\frac{165}x\) we write:$$A=165+x+\frac{3.5\times165}x$$To minimize, we differentiate and set to \(0\):$$\frac{dA}{dx}=1-\frac{3.5\times165}{x^2}\\0=1-\frac{3.5\times165}{x^2}\\\frac{577.5}{x^2}=1\\577.5=x^2\\x=\sqrt{577.5}\approx24.301$$and thus we determine \(y\) as follows.$$y=\frac{165}{\sqrt{165\times3.5}}=\frac{\sqrt{165}}{\sqrt{3.5}}=\sqrt{\frac{330}7}\approx6.861$$

Answer 3

Thus our desired print area's dimensions are \(x\times y\) where \(x,\,y\) are as determined above. For the page itself, add our margins.