Question

Решить задачу : требуется огородить сеткой длиной 600м. зону отдыха прямоугольной формы прилегающую к реке . Определите каковы должны быть длина и ширина участка чтобы он имел наибольшую площадь

Answer 1

Пусть х одна сторона, тогда вторая сторона 300-х. т.к. всего сетки 600м, это периметр, т.е. полупериметр 300 это сумма двух сторон. Составим функцию для площади
S=(300-x)x=300x-x^2
S'=300-2x
300-2x=0
2x=300
x=150 одна сторона,
300-150=150 другая сторона
S=150*150=22500
Р=150+150+150+150=600 м сетки
Ответ 150 и 150

Answer 2

Problem Analysis

To find the dimensions of the rectangular area that will maximize its area, we can use calculus. Let's assume the length of the rectangular area is L and the width is W. We want to maximize the area, which is given by the formula A = L * W. However, we have a constraint that the perimeter of the rectangular area should be 600 meters. The perimeter is given by the formula P = 2L + 2W. We can rewrite this equation as L = (600 - 2W) / 2 and substitute it into the area formula.

Solution

To find the dimensions that maximize the area, we need to find the critical points of the area function. We can do this by taking the derivative of the area function with respect to one of the variables (either L or W), setting it equal to zero, and solving for the variable. Let's take the derivative with respect to L:

A = L * W

Taking the derivative with respect to L:

dA/dL = W

Setting the derivative equal to zero:

W = 0

This means that the width should be zero, which doesn't make sense in this context. Therefore, we need to take the derivative with respect to W:

dA/dW = L

Setting the derivative equal to zero:

L = 0

Again, this means that the length should be zero, which is not possible. Therefore, there are no critical points for the area function.

Since there are no critical points, we need to consider the endpoints of the feasible region. In this case, the feasible region is defined by the constraint that the perimeter should be 600 meters. Let's substitute the expression for L from the perimeter equation:

P = 2L + 2W

600 = 2((600 - 2W) / 2) + 2W

Simplifying the equation:

600 = 600 - 2W + 2W

600 = 600

This equation is true for any value of W. Therefore, the perimeter constraint does not provide any additional information about the dimensions of the rectangular area.

Conclusion

Based on the analysis, there are no critical points for the area function, and the perimeter constraint does not provide any additional information. Therefore, we cannot determine the dimensions of the rectangular area that will maximize its area with the given information.

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