Question

Из цилиндрического стержня изготовили максимальное количество гаек квадратной формы со стороной 12 см с минимальным расходом материала. В каждой гайке проделали отверстие диаметром 6 см. Найдите диаметр стержня. Сколько процентов от объёма стержня составляют отходы при переработке?

Answer 1

Гайка квадратной формы со стороной 12 вписана в стержень. Т.е. диаметр стержня равен диагонали квадрата - 12√2.
Пусть:
S1 - площадь сечения стержня
S2 - площадь квадрата (гайки без дырки)
S3 - площадь отверстия в найке
Тогда:
S1 = π × (12√2 / 2)² = 72π
S2 = 12² = 144
S3 = π × (6/2)² = 9π

S4 (площадь отходов) = S1 - S2 + S3 = 81π - 144
Процент отходов = (81π - 144) / 72π * 100

Answer 2

Problem Analysis

We are given a cylindrical rod from which we need to make the maximum number of square nuts with a side length of 12 cm, using the minimum amount of material. Each nut has a hole with a diameter of 6 cm. We need to find the diameter of the rod and calculate the percentage of waste generated during the process.

Solution

To find the diameter of the rod, we can use the relationship between the side length of the square nut and the diameter of the hole. The diagonal of a square is equal to the side length multiplied by the square root of 2. In this case, the diagonal of the square nut is equal to the diameter of the hole. Therefore, we can set up the following equation:

side length of square nut * sqrt(2) = diameter of hole

Given that the side length of the square nut is 12 cm and the diameter of the hole is 6 cm, we can solve for the diameter of the rod:

12 * sqrt(2) = diameter of rod

Let's calculate the diameter of the rod:

diameter of rod = 12 * sqrt(2) cm

Now, let's calculate the percentage of waste generated during the process. The waste generated is the difference between the volume of the rod and the sum of the volumes of all the nuts. The volume of a cylinder is given by the formula:

volume of cylinder = pi * (radius^2) * height

The volume of a square nut is given by the formula:

volume of square nut = (side length)^2 * height

To calculate the percentage of waste, we can use the following formula:

percentage of waste = (waste volume / rod volume) * 100

Let's calculate the percentage of waste:

1. Calculate the volume of the rod: - The height of the rod is equal to the side length of the square nut, which is 12 cm. - The radius of the rod is half of the diameter, which is (12 * sqrt(2)) / 2 cm. - Using the formula for the volume of a cylinder, we can calculate the volume of the rod.

2. Calculate the volume of a single square nut: - The height of the square nut is equal to the side length of the square nut, which is 12 cm. - Using the formula for the volume of a square nut, we can calculate the volume of a single nut.

3. Calculate the total volume of all the nuts: - Multiply the volume of a single nut by the maximum number of nuts that can be made from the rod.

4. Calculate the waste volume: - Subtract the total volume of all the nuts from the volume of the rod.

5. Calculate the percentage of waste: - Divide the waste volume by the volume of the rod and multiply by 100.

Let's calculate the diameter of the rod and the percentage of waste:

Calculation

1. Diameter of the rod: - side length of square nut = 12 cm - diameter of hole = 6 cm - diameter of rod = 12 * sqrt(2) cm

2. Percentage of waste: - volume of rod = pi * (radius^2) * height - volume of square nut = (side length)^2 * height - total volume of all nuts = volume of square nut * maximum number of nuts - waste volume = volume of rod - total volume of all nuts - percentage of waste = (waste volume / volume of rod) * 100

Let's calculate the values:

1. Diameter of the rod: - side length of square nut = 12 cm - diameter of hole = 6 cm - diameter of rod = 12 * sqrt(2) cm ≈ 16.97 cm

2. Percentage of waste: - volume of rod = pi * (radius^2) * height - volume of square nut = (side length)^2 * height - total volume of all nuts = volume of square nut * maximum number of nuts - waste volume = volume of rod - total volume of all nuts - percentage of waste = (waste volume / volume of rod) * 100

Answer

The diameter of the rod is approximately 16.97 cm. The percentage of waste generated during the process is dependent on the maximum number of nuts that can be made from the rod. Since the maximum number of nuts is not provided in the question, we cannot calculate the exact percentage of waste.