Answer:
[tex]108\ \text{m}^2[/tex]
Step-by-step explanation:
The sides are rectangle is shown in the diagram
The perimeter of the fence will be
[tex]3x+2y=36\\\Rightarrow x=\dfrac{36-2y}{3}[/tex]
Area of the smaller section of the garden
[tex]A=xy\\\Rightarrow A=\dfrac{36-2y}{3}y\\\Rightarrow A=\dfrac{36y-2y^2}{3}[/tex]
Differentiating with respect to y we get
[tex]\dfrac{dA}{dy}=\dfrac{1}{3}(36-4y)[/tex]
Equating with zero we get
[tex]\dfrac{1}{3}(36-4y)=0\\\Rightarrow 36-4y=0\\\Rightarrow y=\dfrac{36}{4}\\\Rightarrow y=9[/tex]
Double derivative of area
[tex]\dfrac{d^2A}{dy^2}=-4<0[/tex]
at y = 9 the area is maximum
[tex]x=\dfrac{36-2y}{3}=\dfrac{36-2\times 9}{3}\\\Rightarrow x=6[/tex]
The sides of the entire garden are [tex]2y=2\times 9=18\ \text{m}[/tex] and [tex]6\ \text{m}[/tex]
The maximum area of the natural fertilizer garden is [tex]18\times 6=108\ \text{m}^2[/tex]
