A Farmer Wants To Build A Rectangular Pen For His Cow, He Wants To Build His Pen Along A River So He Will Only Need To Have Fence Along 3 Of The Sides

A farmer wants to build a rectangular pen for his cow, he wants to build his pen along a river so he will only need to have fence along 3 of the sides. He has 1000 ft of fencing, what should the dimensions be to maximize the area?

How do I solve this problem?

 

Answer:

The max area is 125000 ft² with length = 500 ft and width = 250 ft

Step-by-step explanation:

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Solution in brief:

1. Form an equation with the given information with only 1 variable.

2. Find the first derivative.

3. Find the value of the variable when the first derivative is 0. (This will give us either the max or the min value of the variable)

4. Find the second derivative. If the second derivative is greater than zero, the value found in (3) is the min value. If the second derivative in smaller than zero, the value is the max value.

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STEP 1: Define x and y:

Let the length be x

The width be y

STEP 2: Find the perimeter in term of x and y

Perimeter = length + 2 width

Perimeter = x + 2y

STEP 3: Form the equation for perimeter:

Given that the length of the fencing is 1000 ft

x + 2y = 1000

STEP 4: Make x the subject:

x + 2y = 1000

x = 1000 - 2y

STEP 5: Find the area in term of x and y:

Area = length x width

Area = xy

STEP 6: Find the area in term of y:

Area = xy

Area = (1000 - 2y)y

Area = 1000y - 2y²

STEP 7: Find the first derivative of the area:

A(y) = 1000y - 2y²

A(y) = 1000 - 4y

STEP 8: Find the value of y when A(y) = 0

1000 - 4y = 0

4y = 1000

y = 250

STEP 9: Find the second derivative of the area:

A(y) = 1000 - 4y

A(y) = -4

STEP 10: Test the second derivative

Since A(y) = -4

⇒ A(y) < 0

⇒ A(y) is max when y = 250

STEP 11: Find the value of x when y = 250:

x = 1000 - 2y

x = 1000 - 2(250)

x = 500 ft

STEP 12: Find the max area:

Area = xy

Area = (500)(250)

Area = 125000 ft²

Answer: The max area is 125000 ft² with length = 500 ft and width = 250 ft