Answer :
To solve the problem of finding the radius of a cylinder whose volume is equal to that of a sphere with a radius of 1, and given the cylinder's altitude (height) is 3, we follow these steps:
1. Calculate the volume of the sphere.
The formula for the volume of a sphere is:
[tex]\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \][/tex]
Here, the radius [tex]\( r \)[/tex] of the sphere is given as 1. Plugging in this value, we get:
[tex]\[ V_{\text{sphere}} = \frac{4}{3} \pi (1)^3 = \frac{4}{3} \pi \][/tex]
2. Set up the equation for the volume of the cylinder.
The formula for the volume of a cylinder is:
[tex]\[ V_{\text{cylinder}} = \pi r_{\text{cyl}}^2 h \][/tex]
Here, [tex]\( r_{\text{cyl}} \)[/tex] is the radius of the cylinder, and [tex]\( h \)[/tex] is the height of the cylinder. The height is given as 3. Therefore:
[tex]\[ V_{\text{cylinder}} = \pi r_{\text{cyl}}^2 \cdot 3 \][/tex]
3. Equate the volumes of the sphere and the cylinder.
We know that the volume of the sphere is equal to the volume of the cylinder:
[tex]\[ \frac{4}{3} \pi = \pi r_{\text{cyl}}^2 \cdot 3 \][/tex]
4. Solve for the radius of the cylinder [tex]\( r_{\text{cyl}} \)[/tex].
First, divide both sides of the equation by [tex]\( \pi \)[/tex]:
[tex]\[ \frac{4}{3} = 3 r_{\text{cyl}}^2 \][/tex]
Next, divide both sides by 3 to isolate [tex]\( r_{\text{cyl}}^2 \)[/tex]:
[tex]\[ \frac{4}{3} \div 3 = r_{\text{cyl}}^2 \][/tex]
Simplifying the left-hand side:
[tex]\[ r_{\text{cyl}}^2 = \frac{4}{9} \][/tex]
5. Find the radius [tex]\( r_{\text{cyl}} \)[/tex].
Taking the square root of both sides, we get:
[tex]\[ r_{\text{cyl}} = \sqrt{\frac{4}{9}} = \frac{2}{3} \][/tex]
Therefore, the radius of the cylinder is [tex]\( \frac{2}{3} \)[/tex].
1. Calculate the volume of the sphere.
The formula for the volume of a sphere is:
[tex]\[ V_{\text{sphere}} = \frac{4}{3} \pi r^3 \][/tex]
Here, the radius [tex]\( r \)[/tex] of the sphere is given as 1. Plugging in this value, we get:
[tex]\[ V_{\text{sphere}} = \frac{4}{3} \pi (1)^3 = \frac{4}{3} \pi \][/tex]
2. Set up the equation for the volume of the cylinder.
The formula for the volume of a cylinder is:
[tex]\[ V_{\text{cylinder}} = \pi r_{\text{cyl}}^2 h \][/tex]
Here, [tex]\( r_{\text{cyl}} \)[/tex] is the radius of the cylinder, and [tex]\( h \)[/tex] is the height of the cylinder. The height is given as 3. Therefore:
[tex]\[ V_{\text{cylinder}} = \pi r_{\text{cyl}}^2 \cdot 3 \][/tex]
3. Equate the volumes of the sphere and the cylinder.
We know that the volume of the sphere is equal to the volume of the cylinder:
[tex]\[ \frac{4}{3} \pi = \pi r_{\text{cyl}}^2 \cdot 3 \][/tex]
4. Solve for the radius of the cylinder [tex]\( r_{\text{cyl}} \)[/tex].
First, divide both sides of the equation by [tex]\( \pi \)[/tex]:
[tex]\[ \frac{4}{3} = 3 r_{\text{cyl}}^2 \][/tex]
Next, divide both sides by 3 to isolate [tex]\( r_{\text{cyl}}^2 \)[/tex]:
[tex]\[ \frac{4}{3} \div 3 = r_{\text{cyl}}^2 \][/tex]
Simplifying the left-hand side:
[tex]\[ r_{\text{cyl}}^2 = \frac{4}{9} \][/tex]
5. Find the radius [tex]\( r_{\text{cyl}} \)[/tex].
Taking the square root of both sides, we get:
[tex]\[ r_{\text{cyl}} = \sqrt{\frac{4}{9}} = \frac{2}{3} \][/tex]
Therefore, the radius of the cylinder is [tex]\( \frac{2}{3} \)[/tex].
