A rancher with 750 ft of fencing wants to enclose a rectangu

A rancher with 750 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to (a) Find a function that a models the total area of the four pens. A(w) = 187.5 (b) Find the largest possible total area of the four pens: 67500 ft^2

Solution

Let x = width of the total area
Let y = length of the total area

The total fencing is equal to all the individual fence sections. The top and bottom of the area is 2y and the three dividers on the inside plus the left and right side of the area is 5x.

750 = 2y + 5x
Area = x*y

Now solve the first equation for y and substitute into the area equation.

750 -5x = 2y
y = 375 -2.5x
Area = x *(375 - 2.5x)
Area = 375x - 2.5x^2.

Now find the maximum area (the x-coordinate of the vertex of the graph of the area).

x = -b/(2a)
x = -375/(2*-2.5) = 75

Now that we know the width of the area 75, we can solve for the maximum area (the y-coordinate of the vertex of the area graph).

Max Area = 375(75) - 2.5(75)^2
Max Area = 28125 - 14062.5
Max Area = 14062.5 square ft.