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A rancher wants to create two rectangular pens, as shown in the figure, using an existing fence line as one side. The pens need to have a total area of 972 square feet. What dimensions should be used to minimize the amount of fence used?

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Matheng
A rancher wants to create two rectangular pens, as shown in the attached figure

Let the side parallel to the existing fence line = y
And the sides which is perpendicular to existing fence line = x
The pens need to have a total area of 972 square feet
∴ x y = 972 
∴ y = 972/x

Let the length of needed fence = L
∴ L = y + 3x
substituting with the value of y
∴ [tex]L = \frac{972}{x} +3x[/tex]
differentiating the length with respect to x and equating with zero
∴ [tex] \frac{dL}{dx} = \frac{-972}{x^2} +3 = 0[/tex]
solve for x
∴ x = 18
substituting to find y
∴ y = 54

The dimensions should be used to minimize the amount of fence used
is 18 , 54 feet
Ver imagen Matheng
MrRoyal

The dimensions that minimize the amount of fence is 18 feet by 54 feet

How to determine the dimension that minimizes the pen?

Represent the dimension with x and y.

So, we have:

Area = xy

Perimeter = 3x + y

The area is given as 972.

So, we have:

xy = 972

Make y the subject

y = 972/x

Substitute y = 972/x in P = 3x + y

P = 3x + 972/x

Differentiate

P' = 3 - 972/x^2

Set to 0

3 - 972/x^2 = 0

Rewrite as:

972/x^2  = 3

Multiply through by x^2

3x^2  = 972

Divide by 3

x^2  = 324

Take the square root of both sides

x  = 18

Recall that:

y = 972/x

So, we have:

y = 972/18

Divide

y = 54

Hence, the dimensions that minimize the amount of fence is 18 feet by 54 feet

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