Question

Через образующую цилиндра проведены два взаимно перпендикулярных сечения, площади которых равны 16 и 30 см^2. Найдите боковую поверхность цилиндра

Answer 1

Образующая цилиндра (=высоте цилиндра H) --- общая сторона двух прямоугольников-сечений цилиндра
в плоскости основания --- вписанный в окружность прямой угол, катетами которого являются вторые стороны этих прямоугольников (a и b) 
S1сеч. = aH = 16
S2сеч. = bH = 30
в прямоугольном треугольнике, вписанном в окружность, гипотенуза равна диаметру окружности...
(2R)^2 = a^2 + b^2 ---теорема Пифагора...
4R^2 = 16^2 / H^2 + 30^2 / H^2
R^2 = (16^2+30^2) / (4H^2)
R = V(8^2+15^2) / H
Sбок.цилиндра = 2pi*R*H 
Sбок.цилиндра = 2pi*V(64+225) = 34pi

Answer 2

Problem Analysis

We are given two perpendicular cross-sections of a cylinder, and we need to find the lateral surface area of the cylinder.

Solution

To find the lateral surface area of the cylinder, we can use the formula: Lateral Surface Area = Perimeter of the base * Height

First, we need to find the dimensions of the cylinder using the given areas of the cross-sections.

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

The areas of the cross-sections are given as 16 cm^2 and 30 cm^2. These areas are related to the radius and height of the cylinder.

Using the formula for the area of a circle (A = πr^2), we can find the radius of the cylinder for each cross-section.

For the first cross-section: πr^2 = 16 From this, we can find the radius, r1.

For the second cross-section: πr^2 = 30 From this, we can find the radius, r2.

Next, we need to find the height of the cylinder. The height is the same for both cross-sections.

Now that we have the radius and height, we can calculate the lateral surface area of the cylinder.

Calculation

Let's calculate the lateral surface area of the cylinder using the obtained dimensions.

For the lateral surface area, we use the formula: Lateral Surface Area = 2πrh

Where r is the radius and h is the height of the cylinder.

Conclusion

By following the steps outlined above, we can determine the lateral surface area of the cylinder using the given areas of the cross-sections.