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:
\(\[
V_{\text{sphere}} = \frac{4}{3} \pi r^3
\]\)
Here, the radius \(\( r \)\) of the sphere is given as 1. Plugging in this value, we get:
\(\[
V_{\text{sphere}} = \frac{4}{3} \pi (1)^3 = \frac{4}{3} \pi
\]\)
2. Set up the equation for the volume of the cylinder.
The formula for the volume of a cylinder is:
\(\[
V_{\text{cylinder}} = \pi r_{\text{cyl}}^2 h
\]\)
Here, \(\( r_{\text{cyl}} \)\) is the radius of the cylinder, and \(\( h \)\) is the height of the cylinder. The height is given as 3. Therefore:
\(\[
V_{\text{cylinder}} = \pi r_{\text{cyl}}^2 \cdot 3
\]\)
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:
\(\[
\frac{4}{3} \pi = \pi r_{\text{cyl}}^2 \cdot 3
\]\)
4. Solve for the radius of the cylinder \(\( r_{\text{cyl}} \)\).
First, divide both sides of the equation by \(\( \pi \)\):
\(\[
\frac{4}{3} = 3 r_{\text{cyl}}^2
\]\)
Next, divide both sides by 3 to isolate \(\( r_{\text{cyl}}^2 \)\):
\(\[
\frac{4}{3} \div 3 = r_{\text{cyl}}^2
\]\)
Simplifying the left-hand side:
\(\[
r_{\text{cyl}}^2 = \frac{4}{9}
\]\)
5. Find the radius \(\( r_{\text{cyl}} \)\).
Taking the square root of both sides, we get:
\(\[
r_{\text{cyl}} = \sqrt{\frac{4}{9}} = \frac{2}{3}
\]\)
Therefore, the radius of the cylinder is \(\( \frac{2}{3} \)\).
