MATHEMATICS COLLEGE

farmer ed has 3000 meters of fencing and wants to enclose a rectangle plot that borders on a river. if farmer ed does not fence the side along the river what is the largest area that can be enclosed

Answers

Answer 1

Answer:

Answer:

area = 1500× 750 = $1125000 m^2$

Step-by-step explanation:

we know area of rectangle

for length = l m

and width = b m

$A = lb$

and perimeter

$Perimeter = 2 (length + width)$

but one side length measures is not required because of the river so

He does not use the fence along the side of the river

so we use this formula

Perimeter = P = L + 2 b

Perimeter is 3000 m

$so \ \ 3000 = l +2b$

$l = 3000 - 2b$

so area will be

$A = (3000-2b)b$

it is a quadratic function whose max or min will

occur at the average of the Solutions.

on Solving (3000 - 2b)b = 0

3000 - 2b = 0 or b=0

2b =3000

$b =(3000)/(2) \nb = 1500 m$

or $b = 0 m$

The average of the values are $((0+1500))/(2) = 750$

so for max area we use b= $750 m$

The Length is then L=3000 - 2(750) = 3000 - 1500 = 1500

for max area

length = 1500 m

bredth = 750 m

area = 1500× 750 = $1125000 m^2$

Answer 2

Answer:

Final answer:

The largest area that can be enclosed by Farmer Ed with 3000 meters of fencing along a river (with only three sides fenced) equals 1,125,000 square meters by using principles of mathematical optimization.

Explanation:

In this question, Farmer Ed wants to maximize the area of a rectangle with only three sides fenced, since one side borders on a river. We can use the principles of optimization in mathematics to solve this problem.

With 3000 meters of fencing for three sides, if we denote one side perpendicular to the river as X and the side parallel to the river (which forms the base of the rectangle) as Y, then, the perimeter would be Y+2X which is equal to 3000 meters. So, Y = 3000-2X.

The area A of a rectangle is length times width, or, in this case, A = XY. Substituting Y from the equation above: A = X(3000-2X) = 3000X - 2X^2. To maximize this area, we need to find values of X for which this equation has its maximum value.

The maximum or minimum of a function can be found at points where its derivative is zero. So, we take the derivative of A with respect to X, set it equal to zero, and solve for X.

The derivative, dA/dX is 3000 - 4X. Setting this equal to 0 gives X = 3000/4 = 750. So, the maximum area that Farmer Ed can enclose is when X is 750, and Y is 3000 - 2X = 1500, so the maximum area is 750 * 1500 = 1,125,000 square meters.

Learn more about Mathematical Optimization here:

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Answers

Answer:

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Step-by-step explanation:

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Answers

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