Question

Радиус основания цилиндра равен 6 ,а радиус основания конуса равен 9. Образующая цилиндра равна высоте конуса. Найдите отношение объема конуса к объему цилиндра. напишите объяснения.

Answer 1

V цилиндра = h*pi*6*6
V конуса = 1/3 *h*pi*9*9
Запишем как дробь (1/3 *h*pi*9*9)/(h*pi*6*6)
Сократим (1/3 *9*9)/(6*6) = (3*9)/(6*6) = 9/12 = 0,75

Answer 2

Problem Analysis

We are given the radius of the base of a cylinder (6) and the radius of the base of a cone (9). We are also told that the slant height of the cone is equal to the height of the cylinder. We need to find the ratio of the volume of the cone to the volume of the cylinder.

Solution

To find the ratio of the volume of the cone to the volume of the cylinder, we need to calculate the volumes of both shapes and then divide the volume of the cone by the volume of the cylinder.

Calculating the Volume of the Cone

The volume of a cone can be calculated using the formula: V = (1/3) * π * r^2 * h, where V is the volume, π is a mathematical constant approximately equal to 3.14159, r is the radius of the base, and h is the height or slant height of the cone.

In this case, we are given that the slant height of the cone is equal to the height of the cylinder. Let's assume the height of the cylinder is h. Therefore, the height of the cone is also h.

Using the given radius of the base of the cone (9) and the height of the cone (h), we can calculate the volume of the cone.

Calculating the Volume of the Cylinder

The volume of a cylinder can be calculated using the formula: V = π * r^2 * h, where V is the volume, π is a mathematical constant approximately equal to 3.14159, r is the radius of the base, and h is the height of the cylinder.

In this case, we are given the radius of the base of the cylinder (6) and the height of the cylinder (h). We can calculate the volume of the cylinder using these values.

Finding the Ratio

To find the ratio of the volume of the cone to the volume of the cylinder, we divide the volume of the cone by the volume of the cylinder.

Let's calculate the volumes and find the ratio.

Calculation

Given: - Radius of the base of the cylinder (r_cylinder) = 6 - Radius of the base of the cone (r_cone) = 9 - Height of the cylinder (h_cylinder) = h_cone

Volume of the cone (V_cone) = (1/3) * π * r_cone^2 * h_cone Volume of the cylinder (V_cylinder) = π * r_cylinder^2 * h_cylinder

Substituting the given values: V_cone = (1/3) * π * 9^2 * h_cone V_cylinder = π * 6^2 * h_cylinder

To find the ratio, we divide V_cone by V_cylinder: Ratio = V_cone / V_cylinder

Now, let's calculate the volumes and the ratio.

Calculation Results

Using the given values, we can calculate the volumes and the ratio as follows:

Volume of the cone (V_cone) = (1/3) * π * 9^2 * h_cone Volume of the cylinder (V_cylinder) = π * 6^2 * h_cylinder

Substituting the values: V_cone = (1/3) * π * 81 * h_cone V_cylinder = π * 36 * h_cylinder

Simplifying: V_cone = 27π * h_cone V_cylinder = 36π * h_cylinder

The ratio is given by: Ratio = V_cone / V_cylinder = (27π * h_cone) / (36π * h_cylinder)

Simplifying: Ratio = 27/36 = 3/4

Answer

The ratio of the volume of the cone to the volume of the cylinder is 3:4.

Please let me know if I can help you with anything else.