A poster is to be printed on a rectangular sheet of paper of area \(18 \mathrm{~m}^2\). The margins at the top and bottom of \(75 \mathrm{~cm}\) each and at the sides \(50 \mathrm{~cm}\) each are to be left. Then the dimensions i.e. height and breadth of the sheet, so that the space available for printing is maximum, are respectively.

Let height and breadth of the sheet be ' \(y\) ' \(\mathrm{m}\) and ' \(x\) ' \(\mathrm{m}\) respectively.
\(\therefore x y=180000 \mathrm{~cm}^2\)
\(\therefore y=\frac{180000}{x}\)
\(\therefore \) The area available for printing is
\(\mathrm{A} =(y-150)(x-100) \)
\( =\left(\frac{180000}{x}-150\right)(x-100) \)
\( =180000-\frac{18000000}{x}-150 x-15000 \)
\( =165000-150 x-\frac{18000000}{x} \)
\( \therefore \frac{\mathrm{dA}}{\mathrm{d} x} =0-150+\frac{18000000}{x^2} \)
\( \therefore \frac{\mathrm{dA}}{\mathrm{d} x} =0 \Rightarrow x^2=\frac{18000000}{150}=120000 \)
\( \Rightarrow x=200 \sqrt{3} \mathrm{~cm} \Rightarrow y=\frac{180000}{200 \sqrt{3}}=300 \sqrt{3} \mathrm{~cm}\)
Now, \(\frac{\mathrm{d}^2 \mathrm{~A}}{\mathrm{~d} x^2}=\frac{-36000000}{x^3}\)
\(\therefore \) At \(x=200 \sqrt{3} \mathrm{~cm}, \frac{\mathrm{d}^2 \mathrm{~A}}{\mathrm{~d} x^2} < 0\)
\(\therefore \) Area is maximum at \(x=200 \sqrt{3} \mathrm{~cm}\) and \(\begin{aligned} y & =300 \sqrt{3} \mathrm{~cm} \\ \therefore \quad y & =3 \sqrt{3} \mathrm{~m} \text { and } x=2 \sqrt{3} \mathrm{~m}\end{aligned}\)