A farmer wants to create a rectangular pen, which will be divided into six separate sections, as shown in the accompanying diagram. If he has 600 feet of fencing to use, what outside dimensions will maximize the area of the pen?
1. assign x and y values to the length and width of each box within the pens, and use the formula
600=9x+8y to isolate the y value
(i got the 8 and 9 by counting the number of length and width values existed in the diagram)
you should get y=(600-9x)/8
2. then, since they asked for the maximum area have
f(x)=x((600-9x)/8)
and simplify such that f(x)=75x-(9/8)x^2
3. find the derivative of the function and set it equal to 0
f(x)=75-(18/8)x or 75-(9/4)x
x=100/3
4. then plug the x value into y=(600-9x)/8 to get
y=75/2
5. finally count up the number of x and y values found on on the total length and width (3 for x, 2 for y)
and multiply the numbers by that value
conveniently (75/2)*2=75
and (100/3)*3=100
hope that helps, and i hope that wasn't too late :)
To find the outside dimensions that will maximize the area of the rectangular pen, we need to understand the relationship between the dimensions and the area. Let's set up an equation and then differentiate it to find the maximum area.
Let's assume the length of the rectangular pen is x feet and the width is y feet. We can divide the rectangular pen into six sections, as shown in the diagram.
Now, let's calculate the total fencing needed. There are four vertical sides and five horizontal sides. So, the total fencing required will be:
4x + 5y
Given that the total fencing available is 600 feet, we have:
4x + 5y = 600
Now, we need to express the area of the rectangular pen in terms of x and y. The area of a rectangle is given by:
Area = length * width
So, the area of the rectangular pen is:
Area = (x - 2y) * y
Now, we have two equations:
4x + 5y = 600
Area = (x - 2y) * y
To maximize the area, we need to differentiate the area equation with respect to y and set it equal to 0.
d(Area)/dy = (x - 2y) + (x - 2y) = 0
Simplifying, we get:
2x - 4y = 0
Rearranging this equation, we find:
x = 2y
Now substitute this value of x in the equation 4x + 5y = 600:
4(2y) + 5y = 600
8y + 5y = 600
13y = 600
y = 46.15 (rounded to 2 decimal places)
Now substitute this value of y in x = 2y:
x = 2 * 46.15
x = 92.31
So, the width of the rectangular pen should be approximately 46.15 feet and the length should be approximately 92.31 feet to maximize the area.