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# A poster is to contain $50cm^2$ of matter with borders of 4 cm at top and bottom and of 2 cm on each side. Find the dimensions if the total area of the poster is minimum.

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A)
Toolbox:
• For greatest area, maximise area.
• Working rule for maxima minima.
• Find the function to be maximised or minimised
• Convert the function into single variable
• Apply the conditions for maximum or minimum i.e., first derivative = 0 ( w.r.to the variable converted) and second derivative positive for minima and negative for maxima.
Let the dimensions be $x$ cm length and $y$ cm breadth.
Area of poster = $A = xy$
Given : (x-8)(y-4)=50$or$ y = \large\frac{50}{x-8}+4 \Rightarrow A = \bigg[ \large\frac{50}{x-8}+4 \bigg]^x$Step 3$ \large\frac{dA}{dx}=4-\large\frac{400}{(x-8)^2}=0$Ans$ \Rightarrow x = 18, \: \: y = 9$Step 4$ \large\frac{d^2A}{dx^2}=$positive$ \Rightarrow A\$ is minimum.