# Pam has 90 m of fencing to enclose an area in a petting zoo with two dividers to separate three types

###### Question:

## Answers

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]x0[/tex] as well as [tex]y0[/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[/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 A0[/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}[/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].

[tex]Pam has 90 m of fencing to enclose an area in a petting zoo with two dividers to separate three type[/tex]