Question

Длины сторон прямоугольника натуральные числа, а его периметрравен 4000. Известно, что длинаодной стороны прямоугольника равнаn% от длины другой стороны, где nтакже натуральное число. Какоенаибольшее значение можетпринимать площадь прямоугольника?

Answer 1

Ответ:

Пошаговое объяснение:

a - длина одной стороны

b = a·n / 100  - длина другой стороны

Периметр:

P = 2·(a+b) = 2·(a + a·n/100) = 2a·(1 + n/100)

По условию:

P = 4000

Тогда:

2a·(1 + n/100) = 4000

a = 2000 / (1 + n/100)

b = (2000·n)/ (100+n)

Площадь:

S = a·b

S = 2000·2000·n / ((1 + n/100)·(100 + n))

Находим производную:

S' = 4·10⁸ (100 + n -2n) / (100+n)³

Приравниваем к нулю:

100 + n - 2n = 0

n = 100%

То есть фигура - квадрат.

a = b = 2000 / 2 = 1000

S = a² = 1000² = 1 000 000

Answer 2

Problem Analysis

We are given that the lengths of the sides of a rectangle are natural numbers, and its perimeter is 4000. One side of the rectangle is n% of the length of the other side, where n is also a natural number. We need to determine the maximum possible value for the area of the rectangle.

Solution

To find the maximum possible value for the area of the rectangle, we need to consider the relationship between the sides of the rectangle and the perimeter.

Let's assume that one side of the rectangle is x units. According to the problem statement, the other side of the rectangle is n% of x. This can be expressed as (n/100)x.

The perimeter of the rectangle is given as 4000. The formula for the perimeter of a rectangle is 2(length + width). In this case, the length is x and the width is (n/100)x. Therefore, we can write the equation:

2(x + (n/100)x) = 4000

Simplifying the equation, we get:

2(1 + n/100)x = 4000

Dividing both sides of the equation by 2(1 + n/100), we can solve for x:

x = 4000 / (2(1 + n/100))

Now, we can calculate the area of the rectangle using the formula: area = length * width. Substituting the values of x and (n/100)x, we get:

area = x * (n/100)x = (n/100)x^2

To find the maximum possible value for the area, we need to maximize x and n. Since x and n are natural numbers, we can start by considering the largest possible values for x and n.

The largest possible value for x can be found by setting n = 100 (since n is a natural number). Substituting n = 100 into the equation for x, we get:

x = 4000 / (2(1 + 100/100)) = 4000 / (2(1 + 1)) = 4000 / (2(2)) = 4000 / 4 = 1000

Therefore, the largest possible value for x is 1000.

Now, we can calculate the maximum possible value for the area of the rectangle using the formula:

area = (n/100)x^2 = (100/100)(1000^2) = 1 * 1000000 = 1000000

Therefore, the maximum possible value for the area of the rectangle is 1000000 square units.

Conclusion

The maximum possible value for the area of the rectangle, given that the lengths of the sides are natural numbers and the perimeter is 4000, is 1000000 square units.