Answer:
The greatest area possible is 729[tex]inch^{2}[/tex]
Step-by-step explanation:
i) If a = [tex]-w^{2} + 54w[/tex] where w is the width of the picture frame
differentiating on both sides we get
[tex]\frac{da}{dw} = -2w + 54[/tex]
Differentiating the above again on both sides we get [tex]\frac{d^2a}{dw^2}[/tex] = -2
Since the second order derivative is negative then we can get the solution for the maximum area by equating the fist order derivative equation to 0.
Therefore -2w + 54 = 0 therefore w = 27.
ii) If we substitute w = 27 into the equation in ii) we get
a = [tex]-(27)^2 +(54\times 27)[/tex] = (54 - 27) [tex]\times[/tex](27) = [tex]27^{2}[/tex] = 729
