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ol McDonald would like to build a pin for his favorite pig horse and cow he needs to enclose a rectangular plot of land into three sections with 1200 feet of fence what should the dimensions be such that the area is maximized what is the maximum area of the entire pin

Sagot :

MrRoyal

The area of the pen is the products of its dimensions

  • The dimension of the pen is 300 by 120 feet
  • The maximum area of the pen is 36000 square feet.

Let the dimension of the fence be x by y.

So, we have:

[tex]\mathbf{2x + 5y = 1200}[/tex] --- perimeter

[tex]\mathbf{Area =xy}[/tex] -- area

Subtract 5y from both sides of [tex]\mathbf{2x + 5y = 1200}[/tex]

[tex]\mathbf{2x = 1200 - 5y}[/tex]

Divide both sides by 2

[tex]\mathbf{x = \frac{1200 - 5y}{2}}[/tex]

Substitute [tex]\mathbf{x = \frac{1200 - 5y}{2}}[/tex] in [tex]\mathbf{Area =xy}[/tex]

[tex]\mathbf{Area = \frac{1200 - 5y}{2} \times y}[/tex]

[tex]\mathbf{Area = \frac{1200y - 5y^2}{2}}[/tex]

Split

[tex]\mathbf{Area = 600y - \frac{5}{2}y^2}[/tex]

Differentiate

[tex]\mathbf{A' = 600 -5y}[/tex]

Set to 0

[tex]\mathbf{600 -5y = 0}[/tex]

Add 5y to both sides

[tex]\mathbf{5y = 600}[/tex]

Divide both sides by 5

[tex]\mathbf{y = 120}[/tex]

Substitute [tex]\mathbf{y = 120}[/tex] in [tex]\mathbf{x = \frac{1200 - 5y}{2}}[/tex]

[tex]\mathbf{x = \frac{1200 - 5 \times 120}{2}}[/tex]

[tex]\mathbf{x = \frac{1200 - 600}{2}}[/tex]

[tex]\mathbf{x = \frac{600}{2}}[/tex]

[tex]\mathbf{x = 300}[/tex]

Recall that:

[tex]\mathbf{Area =xy}[/tex]

So, we have:

[tex]\mathbf{Area = 300 \times 120}[/tex]

[tex]\mathbf{Area = 36000}[/tex]

Hence, the maximum area of the pen is 36000 square feet.

Read more about maximum areas at:

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