The total area of a page is `150c m^2dot` The combined width of the margin at the top and bottom is `3c m` and the side 2cm. What must be the dimensions of the page in order that the area of the printed matter may be maximum?

Explanation

Given that The total area of a page is `150c m^2`
The combined width of the margin at the top and bottom is `3c m`
and the side `2cm`.
Let the length and width dimension of total page be `'x'` and be `'y'`.
So, area, `xy=150`
`=>y=150/x`
Dimension of length of printed matter `=x−3`
Dimension of width of printed matter `=y−2`
So, the area of printed matter,`A=(x−3)xx(y−2)`
`A=xy-2x-3y+6`
`=>A=150-2x-3*150/x+6`
`=>A=156-2x-450/x`
differentiating the function and equating it to zero, we get,
`(dA)/(dx)=450/x^2-2 =0`
`=>x^2=225`
`=>x=15 c.m `
hence `y=150/15`
`=10c.m`
again differentiating the function, we get
`(d^2A)/(dx^2)=-900/x^3<0`
Therefore, A is maximum.
so,length of page `=15cm` and width of page `=10cm`.