A farmer has 1000ft of fence and wishes to enclose the largest possible area that has four individual square pens bordered by a rectangular pen of a different width on each end. what are the overall dimensions of the fence area with maximum square footage?
so far i thought i could set up the area equation to be A= 4(x^2) + 4xy and then take the derivative but im not getting the correct answer
what is the Tenths place
what is the tenths place?
IN the fraction 4.5, the five is in the tenths place.
For matt: I don't understand how a rectangular pen can have a different width n each end.
yeah i don't understand the wording but the diagram shows that each end of the entire rectangular pen is 2x
I could be a matter of interpretation.
It said, "four individual square pens bordered by a rectangular pen of a different width on each end"
You have the rectangles fanning out on one side of the squares. Could it not also mean that they fan out on all sides of the squares, so that there would be 10 rectangles with the squares in the middle?
Just wondering if that is the problem.
Anyway, according you your interpretation, I got
20x + 5y = 100
4x + y = 25
y = 25-4x
for
A = 4x^2 + 5x(25-4x)
expanding this, differentiating and setting that equal to zero, gave me
x = 25/6 ft.
If you also got that, try the different interpretation.
To find the overall dimensions of the fence area with maximum square footage, we can use the process of optimization. Let's break down the problem step by step:
Step 1: Define the variables and constraints
Let's assume the width of the rectangular pen on each end is 'x', and the width of each individual square pen is 'y'. We need to maximize the area of the enclosure.
Step 2: Determine the total amount of fence used
We know that the total amount of fence used will be equal to the perimeter of the enclosure. We can set up an equation based on this information:
Perimeter = Total fence used = x + 2y + 2x + 2y + x = 1000 ft
Simplifying the equation, we have:
6x + 4y = 1000
Step 3: Express the area in terms of a single variable
Since we want to maximize the overall area, we need to express it as a function of a single variable. For this, we'll use the equation for the area of four square pens plus a rectangular pen:
Area = 4y^2 + xy
Step 4: Solve for one variable in terms of the other
Now we'll use the perimeter equation (6x + 4y = 1000) to solve for one variable in terms of the other. Let's solve for x in terms of y:
6x = 1000 - 4y
x = (1000 - 4y) / 6
Step 5: Substitute the expression for x in terms of y into the area equation
Now, substitute the expression for x in terms of y into the area equation:
Area = 4y^2 + [(1000 - 4y) / 6] * y
Step 6: Simplify the area equation
To simplify the area equation, you can expand and combine like terms. After simplification, you should end up with a quadratic equation in terms of y.
Step 7: Find the maximum area
To find the maximum area, take the derivative of the area equation with respect to y and set it to zero. Solve for y to find the values that maximize the area.
Once you find the value of y, substitute it back into the perimeter equation to find the corresponding value of x. These values will give you the overall dimensions of the fence area with maximum square footage.