A rectangular page is to contain 30 sqaure inches of print. The margins on each side are 1 inch. Find the dimensions of the page sucha that the least amount of paper is used.

Question

A farmer plans to fence a rectangular pasture adjacent to a river. The pasture must contain 245,000 square meters in order to provide enough grass for the herd. No fencing is needed along the river. what dimensions will require the least amount of fencing?

Hello I am really struggling with making the equation for each I then understand after making the equation to take the derviaitve and critical numbers then plugging in. Can you please help me figure out a way to find equations thank you

Answer 1

The page has a print area with width x and height y. Since xy=30, y = 30/x

The page dimensions including borders are x+2 and y+2.

You want minimal page area. So,

a = (x+2)(y+2) = (x+2)(30/x + 2) = 2x + 34 + 60/x
now just find x where da/dx = 0

The fencing problem is similar. xy=245000
If the x dimension is along the river, then the amount of fencing needed is

f = x+2y = x+2(245000/x)
now find x such that df/dx=0