You are designing a rectangular dog pen off the back of your house. The House will form one side of the dog pen. The pen is to have an area of 800 ft.² but you want to minimize the amount of fencing that you need for the three sides of the pen not against the house. This will include an opening that it's 4 feet wide where it will be placed. How many feet of fencing do you need to build the pen?

To find out how many feet of fencing you need to build the pen, you can follow these steps:

1. Calculate the dimensions of the pen: Since the area is given as 800 ft² and one side is formed by the house, the pen will have two sides of length 'a' and 'b'. This gives us the equation: a * b = 800.

2. Consider the fencing required for the three sides not against the house: To minimize the amount of fencing, we need to find the minimum perimeter of the pen. Since one side is formed by the house, we only need to calculate the fencing required for the other two sides.

3. Determine the dimensions of the pen's opening: Since there is an opening of 4 feet wide, we need to subtract this width from the length of one of the sides in order to calculate the other side.

4. Calculate the length of one side of the pen: Let's assume the length of one side is 'a'. Since the opening is 4 feet wide, the length of the other side will be 'a-4'.

5. Calculate the possible values of 'a': From equation 1, we know that a * b = 800. Substituting the value of 'b' (a-4) into the equation, we get a(a-4) = 800.

6. Determine the value of 'a' and 'b': Solve the quadratic equation a(a-4) = 800 to find the possible values of 'a'. The positive value of 'a' will give us the length of one side, and the corresponding value of 'b' can be calculated as (a-4).

7. Calculate the fencing required for the two sides: The fencing required for the two sides of length 'a' and 'b' will be: 2 * (a + b).

8. Add the fence requirement for the opening: Since the opening is 4 feet wide, we need to add 4 feet to the total perimeter.

By following these steps, you will be able to determine the amount of fencing you need to build the dog pen.

To minimize the amount of fencing needed for the three sides of the pen not against the house, we need to determine the dimensions of the pen.

Let's assume the length of the pen is 'x' feet and the width is 'y' feet.

Given that the area of the pen is 800 ft², we have the equation: x * y = 800.

Since the house forms one side of the pen, we only need two additional sides of fencing.

The total fencing needed is the sum of the length, the width, and the opening for the entrance.

The length of fencing needed is x + y + 4 (since the opening is 4 feet wide).

Now, let's solve for 'x' and 'y' to find the dimensions and then calculate the total fencing needed.

1. Solve the equation: x * y = 800 for 'y':
y = 800 / x

2. Substitute the value of 'y' in the equation: Total fencing = x + y + 4:
Total fencing = x + (800 / x) + 4

To find the minimum fencing, we need to minimize the equation for total fencing. We can find the minimum value by taking the derivative of the equation and setting it equal to zero.

3. Take the derivative of the equation: Total fencing = x + (800 / x) + 4 with respect to 'x':
d(Total fencing) / dx = 1 - (800 / x²)

4. Set the derivative equal to zero and solve for 'x':
1 - (800 / x²) = 0
(800 / x²) = 1
800 = x²
x = √800
x ≈ 28.28 feet (rounded to the nearest hundredth)

5. Substitute the value of 'x' back into the equation: y = 800 / x
y = 800 / 28.28
y ≈ 28.28 feet

Now, we have the dimensions of the rectangular pen: length = x ≈ 28.28 feet and width = y ≈ 28.28 feet.

6. Calculate the total fencing needed:
Total fencing = x + y + 4
≈ 28.28 + 28.28 + 4
≈ 60.56 feet (rounded to the nearest hundredth)

Therefore, you would need approximately 60.56 feet of fencing to build the pen.

xy = 800

f = x+2y-4

To minimize p, substitute in for x or y, and find where p'=0.

Check to be sure it does not matter where the gate is. (on the x side or the y side)