Question

Диагональ осевого сечения цилиндра равна 24корень из 3 см и наклонена к плоскости его основания под углом 30 найдите площадь боковой поверхности цилиндра

Answer 1

До конца досчитаешь сам, там чисто посчитать

Answer 2

Calculation of the Cylinder's Lateral Surface Area

To find the lateral surface area of a cylinder, we need to know the radius of the base and the height of the cylinder. In this case, we are given the diagonal of the cross-section of the cylinder and the angle at which it is inclined to the plane of its base.

Let's break down the problem step by step:

1. We are given that the diagonal of the cross-section of the cylinder is equal to 24√3 cm. This diagonal represents the hypotenuse of a right triangle formed by the radius of the base, the height of the cylinder, and the diagonal itself.

2. The diagonal is inclined to the plane of the base at an angle of 30 degrees.

To find the radius and height of the cylinder, we can use trigonometric ratios.

Let's denote the radius of the base as r and the height of the cylinder as h.

Using trigonometry, we can determine the following relationships:

- The sine of the angle between the diagonal and the height is equal to the ratio of the height to the diagonal: sin(30) = h / (24√3). - The cosine of the angle between the diagonal and the radius is equal to the ratio of the radius to the diagonal: cos(30) = r / (24√3).

Simplifying these equations, we get:

- h = (24√3) * sin(30). - r = (24√3) * cos(30).

Now, we can calculate the values of h and r:

- h = (24√3) * sin(30) = 12 cm. - r = (24√3) * cos(30) = 20√3 cm.

The lateral surface area of a cylinder can be calculated using the formula:

Lateral Surface Area = 2 * π * r * h.

Substituting the values we found:

Lateral Surface Area = 2 * π * (20√3) * 12.

Calculating this expression, we get:

Lateral Surface Area ≈ 1440π cm².

Therefore, the lateral surface area of the cylinder is approximately 1440π cm².

Please note that the value of π is an irrational number and can be approximated as 3.14 for most calculations.

Let me know if there's anything else I can help you with!