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A farmer has 300 meters of fencing to create a rectangular pen divided into two equal sections. What dimensions maximize the total area of the pen?​

Sagot :

chendayao

Answer:

my ans.

Step-by-step explanation:

To maximize the total area of the rectangular pen divided into two equal sections using 300 meters of fencing, we can follow these steps:

Let's denote:

- x as the width of the pen

- y as the length of each divided section

Given that the total fencing is 300 meters, the perimeter of the pen can be expressed as:

2x + 3y = 300

x + 1.5y = 150

x = 150 - 1.5y

The total area (A) of the pen can be calculated as the product of the width and the sum of the lengths of both sections:

A = x \times 2y

A = (150 - 1.5y) \times 2y

A = 300y - 3y^2

To find the dimensions that maximize the total area, we need to find the critical points of the area function. Let's differentiate the area function with respect to y and set it to zero to find the critical points:

\frac{dA}{dy} = 300 - 6y

Setting the derivative to zero:

300 - 6y = 0

6y = 300

y = 50

Substitute the value of y back into the equation for the width (x = 150 - 1.5y):

x = 150 - 1.5(50)

x = 75

Therefore, the dimensions that maximize the total area of the pen are:

- Width (x): 75 meters

- Length of each divided section (y): 50 meters

This configuration will result in the maximum area for the rectangular pen divided into two equal sections using 300 meters of fencing.

poisonedren

STEP-BY-STEP SOLUTION

To maximize the area of a rectangular pen divided into two equal sections using 300 meters of fencing, we need to find the dimensions that will give us the largest area.

Let the length of the pen is x meters, and the width will be y meters. Since the pen is divided into two equal sections, we can set up the following equation to represent the total perimeter of the pen:

  • 2(x + y) = 300

Solving for x, we get:

  • x + 150

Now, we can find the area of the pen by multiplying the length and the width:

  • Area = x * y

Substituting the equation we found for x, we get:

  • Area = (150 - y) * y

To find the maximum area, we need to find the value of y that will give us the largest area. To, we can take the derivative of the area function with respect to y, set it equal to zero, and solve for y.

  • Area' = 2 * (150 - y)

Setting Area' equal to zero, we get:

  • 2 * (150 - y) = 0

Solving for y, we get:

  • 150 - y = 0
  • y = 150

Now that we have found the y, we can substitute it back into the equation we found for x:

  • x + 150 = 150
  • x = 0

However, since the length cannot be zero, we need to adjust the width slightly.

Since the pen is divided into two equal sections, we can set the length to be slightly larger than zero, say 1 meter, to make sure we have a valid rectangle. Substituting x = 1 into the equation we found for y, we get:

  • y = 149

So, the dimensions that will maximize the total area of the pen are a length of 1 meter and a width of 149 meters.

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