A farmer wants to fence an area of 13.5 million square feet in a rectangular field and then divide it in half with a fence paral

Question

2 years ago

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A farmer wants to fence an area of 13.5 million square feet in a rectangular field and then divide it in half with a fence paral

lel to one of the sides of the rectangle. Let y represent the length (in feet) of a side perpendicular to the dividing fence, and let x represent the length (in feet) of a side parallel to the dividing fence. Let F represent the length of fencing in feet. Write an equation that represents F in terms of the variable x.

Mathematics

1 answer:

denis23 [38] · Answer 1 · 2023-08-25T17:38:30+03:00

Answer:

F = 3x +(2.7×10^7)/x

Step-by-step explanation:

The formulas for area and perimeter of a rectangle can be used to find the desired function.

The area of the rectangle will be the product of its dimensions:

A = LW

Using the given values, we have ...

13.5×10^6 = xy

Solving for y gives ...

y = (13.5×10^6)/x

<h3>Perimeter</h3>

The perimeter of the rectangle is the sum of the side lengths:

P = 2(L+W) = 2(x+y)

<h3>Fence length</h3>

The total amount of fence required is the perimeter plus one more section that is x feet long.

F = 2(x +y) +x = 3x +2y

Substituting for y, we have a function of x:

F = 3x +(2.7×10^7)/x

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<em>Additional comment</em>

The length of fence required is minimized for x=3000. The overall size of that fenced area is x=3000 ft by y=4500 ft. Each half is 3000 ft by 2250 ft. Half of the total 18000 ft of fence is used for each of the perpendicular directions: 3x=2y=9000 ft.