Question

Радиус основания цилиндра равен 17,а его образующая равна 15. Параллельно оси цилиндра проведена плоскость альфа так, что площадь сечения цилиндра плоскостью альфа равна 240. Найдите расстояние от оси цилиндра до плоскости альфа. ВСЕ С РЕШЕНИЕМ И ХОРОШИМ РЕСУРСОМ.

Answer 1

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Answer 2

Problem Analysis

We are given the radius of the base of a cylinder (17) and its height (15). A plane is parallel to the axis of the cylinder and intersects it, creating a cross-sectional area of 240. We need to find the distance from the axis of the cylinder to the plane.

Solution

To solve this problem, we can use the formula for the area of a circle, which is A = πr^2, where A is the area and r is the radius. We can also use the formula for the volume of a cylinder, which is V = πr^2h, where V is the volume, r is the radius, and h is the height.

First, let's find the area of the base of the cylinder using the given radius: A_base = π(17^2) = 289π.

Next, let's find the volume of the cylinder using the given radius and height: V_cylinder = π(17^2)(15) = 4335π.

Now, let's find the height of the cross-sectional area created by the plane. Since the area of the cross-section is given as 240, we can set up the following equation: 240 = A_base - A_section, where A_section is the area of the cross-section.

Rearranging the equation, we get: A_section = A_base - 240.

Substituting the value of A_base, we have: A_section = 289π - 240.

To find the radius of the cross-sectional area, we can rearrange the formula for the area of a circle: A_section = πr_section^2.

Substituting the value of A_section, we have: 289π - 240 = πr_section^2.

Simplifying the equation, we get: πr_section^2 = 289π - 240.

Dividing both sides of the equation by π, we have: r_section^2 = 289 - 240/π.

Taking the square root of both sides of the equation, we get: r_section = √(289 - 240/π).

Now, let's find the height of the cross-sectional area. Since the cross-sectional area is a circle, the height is equal to the diameter of the circle. The diameter is twice the radius, so: d_section = 2r_section = 2√(289 - 240/π).

Finally, let's find the distance from the axis of the cylinder to the plane. This distance is equal to the radius of the cylinder minus the radius of the cross-sectional area: distance = 17 - r_section.

Substituting the value of r_section, we have: distance = 17 - √(289 - 240/π).

Therefore, the distance from the axis of the cylinder to the plane is 17 - √(289 - 240/π).

Calculation

Let's calculate the value of the distance using the given values.

The value of π is approximately 3.14159.

Substituting the values into the equation, we have: distance = 17 - √(289 - 240/π) = 17 - √(289 - 76.394).

Calculating the value inside the square root, we have: distance = 17 - √(212.606) = 17 - 14.586 = 2.414.

Therefore, the distance from the axis of the cylinder to the plane is approximately 2.414.

Answer

The distance from the axis of the cylinder to the plane is approximately 2.414.