Let W represent width of rectangle and L represent length pf rectangle.
We have been given that a rancher with 750 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle.
First of all, we need to draw a diagram using our given information as shown in the attachment.
We can see that the total fencing will be equal to perimeter of four pens.
[tex]\text{Perimeter}=2L+5W[/tex]
Now we will equate total fencing with perimeter as:
[tex]2L+5W=750[/tex]
Let us solve for L.
[tex]2L+5W-5W=750-5W[/tex]
[tex]2L=750-5W[/tex]
[tex]\frac{2L}{2}=\frac{750-5W}{2}[/tex]
[tex]L=\frac{750-5W}{2}[/tex]
We know that the area of rectangle is length times width.
[tex]\text{Area}=W\cdot L[/tex]
Upon substituting value of L in area equation, we will get:
[tex]\text{Area}=W\cdot \bigg(\frac{750-5W}{2}\bigg)[/tex]
[tex]A(W)=\frac{W\cdot 750-W\cdot 5W}{2}[/tex]
[tex]A(W)=\frac{750W-5W^2}{2}[/tex]
[tex]A(W)=375W-2.5W^2[/tex]
Therefore, our required function would be [tex]A(W)=375W-2.5W^2[/tex].
Using the area and perimeter of triangle formula, the function which models the total area of the four pens is A(L) = 0.4L² + 75L
Recall :
Dividing the pens into 4 whuvhbarw parallel means the The perimeter of of the fence can be defined thus :
5 widths and 2 lengths
Perimeter = 2(L + W)
750 = 2(2L + 5W)
750 = 4L + 10W
We can solve for Width, W ;
10W = 750 - 4L
Divide both sides by 10
W = 75 - 0.4L
Recall ;
Area of rectangle = Length × width = L × W
Substitute, W = 75 - 0.4L into the area Formula ;
A(L) = L × (75 - 0.4L)
A(L) = 75L - 0.4L²
A(L) = 0.4L² + 75L
Therefore, the function which models the total area of the four pens is A(L) = 0.4L² + 75L
Learn more :https://brainly.com/question/18796573
