How Can I Maximize the Area of a Window Using These Equations?

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The discussion focuses on maximizing the area of a window using the equations for area \(A\) and perimeter \(P\). The relevant equations are \(A=\dfrac{1}{2}\pi r^2 + 2rh\) and \(P=2h+2r+\pi r=12+3\pi\). The solution involves expressing \(h\) in terms of \(r\) from the perimeter equation, substituting it into the area equation, differentiating the resulting equation with respect to \(r\), and solving for \(r\) to find the optimal dimensions for maximum area.

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Does anyone know how to work this out. Any help much appreciated. it is question 32 I'm stuck on
 

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Two equations are relevant here:

$$A=\dfrac12\pi r^2+2rh\quad(1)$$

and

$$P=2h+2r+\pi r=2h+(2+\pi)r=12+3\pi\quad(2)$$

Now proceed as follows:

a) solve $(2)$ for $h$ in terms of $r$ and substitute that for $h$ in $(1)$, giving $(1*)$.

b) differentiate $(1*)$ with respect to $r$, equate the result to $0$ and solve for $r$.

c) state $2r$ in meters as the width of the window with maximum area.
 

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