Question

В полушар вписан цилиндр,причем одно из оснований цилиндра лежит в плоскости диаметрального круга полушара,а высота цилиндра втрое меньше радиуса полушара.Найдите отношение объема цилиндра к объему полушара. С решением пожалуйста!

Answer 1

Из прямоугольного треугольника R^2=h^2+r^2;
R^2=(R/3)^2+r^2;  r^2=R^2-R^2/9; 9r^2=9R^2-R^2; r^2=8R^2/9
r=2√2R/3; 
h=R/3
объем цилиндра V1=pir^2=8piR^2/9*(R/3)=8piR^3/27
объем полушария V2=4piR^3/6
V1/V2=(8piR^3/27)/(4piR^3/3=6)=4/9

Answer 2

Problem Analysis

We are given a hemisphere with a cylinder inscribed inside it. One of the bases of the cylinder lies in the plane of the hemisphere's diameter, and the height of the cylinder is one-third of the hemisphere's radius. We need to find the ratio of the volume of the cylinder to the volume of the hemisphere.

Solution

Let's assume the radius of the hemisphere is R.

The height of the cylinder is one-third of the hemisphere's radius, so the height of the cylinder is R/3.

The volume of a cylinder is given by the formula: V_cylinder = π * r^2 * h, where r is the radius of the cylinder and h is the height of the cylinder.

The volume of a hemisphere is given by the formula: V_hemisphere = (2/3) * π * R^3.

To find the ratio of the volume of the cylinder to the volume of the hemisphere, we need to divide the volume of the cylinder by the volume of the hemisphere:

Ratio = V_cylinder / V_hemisphere

Let's substitute the values and calculate the ratio.

Calculation

Given: - Radius of the hemisphere (R) = R - Height of the cylinder (h) = R/3

Volume of the cylinder (V_cylinder) = π * r^2 * h

Volume of the hemisphere (V_hemisphere) = (2/3) * π * R^3

Substituting the values: V_cylinder = π * (R/2)^2 * (R/3) = (π/12) * R^3

V_hemisphere = (2/3) * π * R^3

Ratio = V_cylinder / V_hemisphere = [(π/12) * R^3] / [(2/3) * π * R^3]

Simplifying the expression: Ratio = (π/12) * R^3 / (2/3) * π * R^3 = (π/12) * R^3 * (3/2) * π * R^3

Canceling out the common terms: Ratio = 1/8

Answer

The ratio of the volume of the cylinder to the volume of the hemisphere is 1/8.

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