Respuesta :
Given:
An advertisement consists of a rectangular printed region plus 3-cm margins on the sides and 5-cm
margins at top and bottom.
The area of the printed region is to be 162 cm².
Required:
To find the dimensions of the printed region that minimize the total area.
Explanation:
Let the length and width of the printed region be l and w.
Then it is given that the printed area
[tex]A1=lw[/tex]So
[tex]\begin{gathered} lw=162cm^2 \\ \\ l=162w^{-1} \end{gathered}[/tex]Now, the total length is (l+10) cm and the total width is (w+6) cm.
So the total area A is
[tex]\begin{gathered} A=(l+10)(w+6) \\ \\ A=(162w^{-1}+10)(w+6) \\ \\ A=162+972w^{-1}+10w+60 \\ \\ A=222+972w^{-1}+10w \end{gathered}[/tex]To find the maximum area, we find the extreme points of A by differentiating A with respect to w and setting the resulting derivative to zero. So,
[tex]\frac{dA}{dw}=-972w^{-2}+10[/tex][tex]\begin{gathered} 0=972w^{-2}+10 \\ \\ 972w^{-2}=10 \\ \\ 972=10w^2 \\ \\ w^2=\frac{972}{10} \\ \\ w^2=97.2 \\ \\ w=\sqrt{97.2} \\ \\ w=9.8590cm \end{gathered}[/tex]And the length is
[tex]\begin{gathered} l=\frac{162}{9.8590} \\ \\ l=16.4316cm \end{gathered}[/tex]Therefore, to minimize the total area the length should be 16.4 cm and the width should be 9.9 cm.
Final Answer:
To minimize the total area the length should be 16.4 cm and the width should be 9.9 cm.
