Answer:
The area function is
[tex]A=\frac{135}{2}x-\frac{9}{2}x^2[/tex].
The domain and range of A is [tex](0,15m)[/tex] and [tex](0, 253.125 m^2][/tex].
Step-by-step explanation:
The given length of fencing is [tex]90 m[/tex].
Let the length and width of each pen be [tex]x[/tex] and [tex]y[/tex] respectively as shown in the figure.
As there are 3 pens, so, the total area,
[tex]A= 3 xy \;\cdots (i)[/tex]
From the figure the total length of fencing is [tex]6x+4y[/tex].
Here, for a significant area for the animals, [tex]x>0[/tex] as well as [tex]y>0[/tex] as [tex]x[/tex] and [tex]y[/tex] are the sides of ben.
From the given value:
[tex]6x+4y=90\;\cdots (ii)[/tex]
[tex]\Rightarrow y=\frac {45}{2}-\frac{3x}{2}[/tex]
Now, from equation (i)
[tex]A=3x\left(\frac {45}{2}-\frac{3x}{2}\right)[/tex]
[tex]\Rightarrow A=\frac{135}{2}x-\frac{9}{2}x^2\;\cdots (iii)[/tex]
This is the required area function in the terms of variable [tex]x[/tex].
For the domain of area function, from equation (ii)
[tex]x=15-\frac{2y}{3}[/tex]
[tex]\Rightarrow x<15 m[/tex] [as y>0]
So, the domain of area function is [tex](0,15m)[/tex].
For the range of area function:
As [tex]x \rightarrow 0[/tex] or [tex]y\rightarrow 0[/tex], then [tex]A\rightarrow 0[/tex] [from equation (i)]
[tex]\Rightarrow A>0[/tex]
Now, differentiate the area function with respect to [tex]x[/tex] .
[tex]\frac {dA}{dx}=\frac{135}{2}-9x[/tex]
Equate [tex]\frac {dA}{dx}[/tex] to zero to get the extremum point.
[tex]\frac {dA}{dx}=0[/tex]
[tex]\Rightarrow \frac{135}{2}-9x=0[/tex]
[tex]\Rightarrow x=\frac{15}{2}[/tex]
Check this point by double differentiation
[tex]\frac {d^2A}{dx^2}=-9[/tex]
As, [tex]\frac {d^2A}{dx^2}<0[/tex], so, point [tex]x=\frac{15}{2}[/tex] is corresponding to maxima.
Put this value back to equation (iii) to get the maximum value of area function. We have
[tex]A=\frac{135}{2}\times \frac {15}{2}-\frac{9}{2}\times \left(\frac {15}{2}\right)^2[/tex]
[tex]\Rightarrow A=253.125 m^2[/tex]
Hence, the range of area function is [tex](0, 253.125 m^2][/tex].
