Step 1: Write down the formulas for the volume of a cone and a sphere.
Volume of a cone (\( V_{\text{cone}} \)) = \(\frac{1}{3} \pi r^2 h\)
Volume of a sphere (\( V_{\text{sphere}} \)) = \(\frac{4}{3} \pi r^3\)
where \( r \) is the radius and \( h \) is the height.
Step 2: Use the given condition that their volumes are the same.
Given that \( V_{\text{cone}} = V_{\text{sphere}} \):
\[
\frac{1}{3} \pi r^2 h = \frac{4}{3} \pi r^3
\]
Step 3: Simplify the equation to find the ratio \( r : h \).
First, cancel common terms on both sides. Both sides have \(\frac{1}{3}\) and \(\pi\). Also, both sides have \( r^2 \):
\[
r^2 h = 4 r^3
\]
Divide both sides by \( r^2 \) (assuming \( r \neq 0 \)):
\[
h = 4r
\]
Now, find the ratio \( r : h \).
Divide both sides by \( h \):
\[
1 = 4 \frac{r}{h}
\]
Divide both sides by 4:
\[
\frac{1}{4} = \frac{r}{h}
\]
So, the ratio \( r : h \) is \( 1 : 4 \).
Step 4: Final Answer.
The ratio \( r : h \) is \( 1 : 4 \).
\[
(2) 1:4
\]