Question

В цилиндр вписан куб. Найдите отношение площади полной поверхности цилиндра к площади поверхности куба

Answer 1

Площадь цилиндра есть сумма площади боковой поверхности+2 площади основания.
Площадь основания πR², где R - радиус основания.
Площадь боковой поверхности h·L=2πR·h, где h - высота, L - длина окружности основания.
Поскольку в цилиндр вписан куб, то высота цилиндра h равна длине ребра куба. Ребро куба равно √2·R. Диагональ основания куба есть диаметр окружности основания цилиндра,т.е. 2R. Зная диагональ, мы можем найти сторону основания куба а, она же его ребро. а=2R·√2/2=√2·R.
Площадь поверхности куба равна Sк=6а²=6·2R²=12R²
Общая площадь цилиндра равна Sц=2πR²+2πR·h=2πR²+2πR·√2R=2πR²(1+√2)
Sц/Sк=π(1+√2)/6

Answer 2

Finding the Ratio of the Surface Area of a Cylinder to the Surface Area of a Cube

To find the ratio of the surface area of a cylinder to the surface area of a cube, we need to calculate the surface area of both shapes and then divide the surface area of the cylinder by the surface area of the cube.

Let's assume that the side length of the cube is s and the height of the cylinder is h.

Surface Area of a Cylinder

The surface area of a cylinder consists of two circular bases and the lateral surface area. The formula for the surface area of a cylinder is:

Surface Area of Cylinder = 2πr² + 2πrh

Where: - r is the radius of the circular base of the cylinder - h is the height of the cylinder

Surface Area of a Cube

The surface area of a cube is the sum of the areas of its six faces. Since all faces of a cube are congruent squares, the formula for the surface area of a cube is:

Surface Area of Cube = 6s²

Where: - s is the side length of the cube

Calculating the Ratio

To find the ratio of the surface area of the cylinder to the surface area of the cube, we divide the surface area of the cylinder by the surface area of the cube:

Ratio = (Surface Area of Cylinder) / (Surface Area of Cube)

Now, let's calculate the ratio using the given formulas.

Calculation

From the given information, we don't have the specific values for the radius of the cylinder or the side length of the cube. Therefore, we cannot calculate the exact ratio without these values.

However, we can provide the general formula and explain the steps to calculate the ratio once the specific values are known.

To calculate the ratio, you need to know the values of r (radius of the cylinder) and s (side length of the cube). Once you have these values, you can substitute them into the formulas for the surface area of the cylinder and the surface area of the cube, and then divide the surface area of the cylinder by the surface area of the cube to find the ratio.

Let me know if you have the specific values for the radius of the cylinder and the side length of the cube, and I can help you calculate the ratio.