The farmer wants to use the 500 m to an enclosure divided into two equal areas, What is the total maximum area that can is achieved with the 500 m fence. using differential calculus to the solution.
Enclosure is divide into two equal parts.
Thats why you hawe width 3 times.
Two times of end of fence and one time in midle.
You hawe length 2 times both times of end of fence.
So:
3 W + 2 L = 500
3 W = 500 - 2 L = 2 ( 250 - L )
3 W = 2 ( 250 - L ) Divide both sides by 3
W = ( 2 / 3 ) ( 250 - L )
A = W * L
A = ( 2 / 3 ) ( 250 - L ) * L
A = ( 2 / 3 ) ( 250 L - L ^ 2 )
dA / dL = 250 - 2 L = o
250 - 2 L = 0
250 = 2 L Divide both sides by 2
125 = L
L = 125 m
W = ( 2 / 3 ) ( 250 - L )
W = ( 2 / 3 ) ( 250 - 125 )
W = ( 2 / 3 ) * 125
W = 250 / 3 m
A = W * L
A = 125 * 250 / 3 = 31250 / 3 =
10416.66667 m ^ 2
P.S.
Izvini zbog mog lošeg engleskog.
hvala ti puuuuuuuno, I tvoj engleski je suoper
To find the total maximum area of the enclosure using differential calculus, we can follow these steps:
1. Identify the variables: Let's denote the length of one side of the rectangular enclosure as "x" and the width of the other side as "y". We are given that the perimeter of the enclosure is 500m, so we have the equation: 2x + 2y = 500.
2. Express one variable in terms of the other: We can rearrange the equation above to express y in terms of x: y = (500 - 2x) / 2.
3. Formulate the area in terms of one variable: The area of a rectangle is given by A = x * y. Substitute the expression for y from step 2 into the area equation: A = x * (500 - 2x) / 2.
4. Simplify the area equation: Multiply x by 500 - 2x and divide the result by 2: A = (500x - 2x^2) / 2. Simplify this expression further: A = 250x - x^2.
5. Find the critical points: To find the maximum area, we need to find the values of x where the derivative of the area equation is zero. Differentiate the area equation with respect to x: A' = 250 - 2x. Set this equal to zero and solve for x: 250 - 2x = 0. Solve for x: x = 125.
6. Determine the maximum area: Substitute the value of x into the area equation to find the corresponding y: y = (500 - 2(125)) / 2 = 125. Calculate the area by substituting the values of x and y: A = 125 * 125 = 15,625 square meters.
Therefore, the total maximum area that can be achieved with the 500m fence is 15,625 square meters.