Need to make a rectangular pen for pigs that will enclose a total area of 169 square feet. What is the least amount of fencing that will be needed?

The least amount of fencing needed for the rectangular pen is 72.19 feet.

Step-by-step explanation:

The area and perimeter equations of the pen are, respectively:

(1)

(2)

Where:

- Perimeter, in feet.

- Area, in square feet.

- Width, in feet.

- Length, in feet.

Let suppose that total area is known and perimeter must be minimum, then we have a system of two equations with two variables, which is solvable:

From (2):

(2) in (1):

And the first and second derivatives of the expression are, respectively:

(3)

(4)

Then, we perform the First and Second Derivative Test to the function:

First Derivative Test

Given that dimensions of the rectangular pen must positive nonzero variables:

Second Derivative Test

In a nutshell, the critical value for the width of the pen leads to a minimum perimeter.

If we know that , then the value of the perimeter of the rectangular pen is:

By (2):

Lastly, by (1):

The least amount of fencing needed for the rectangular pen is 72.19 feet.

Step-by-step explanation:

To evaluate this expression, let's first plug in the value of m into the expression.

1 + (- ) - (- )

To make this expression easier to solve, let's convert every value into a fraction with a a common denominator.

6/6 + (-4/6) - (-27/6)

Now we can solve.

When you add a negative number, it's the same thing as subtracting, so:

6/6 + (-4/6) can be written as: 6/6 -4/6, which is equal to 2/6

Now our expression is: 2/6 - (-27/6)

When you subtract a negative number, it's the same thing as adding that number, so:

2/6 - (-27/6) can be written as: 2/6 + 27/6

2/6 + 27/6 = 29/6

Since 29/6 is already in it's most simplified form, 29/6 is your answer.