Let's consider the initial volume of the cone and how it changes when the radius is multiplied by 3 while the height remains the same.
1. Initial Volume Calculation:
The volume [tex]\( V \)[/tex] of a cone with radius [tex]\( r \)[/tex] and height [tex]\( h \)[/tex] is given by the formula:
[tex]\[
V = \frac{1}{3} \pi r^2 h
\][/tex]
2. Initial Volume:
When the radius is [tex]\( r \)[/tex] and the height is [tex]\( h \)[/tex], the volume [tex]\( V_1 \)[/tex] can be represented as:
[tex]\[
V_1 = \frac{1}{3} \pi r^2 h
\][/tex]
3. New Radius:
The radius is multiplied by 3, so the new radius is [tex]\( 3r \)[/tex].
4. New Volume Calculation:
Using the same height [tex]\( h \)[/tex] and the new radius [tex]\( 3r \)[/tex], the new volume [tex]\( V_2 \)[/tex] is:
[tex]\[
V_2 = \frac{1}{3} \pi (3r)^2 h
\][/tex]
5. Simplify the New Volume:
We can simplify the expression for [tex]\( V_2 \)[/tex] as follows:
[tex]\[
V_2 = \frac{1}{3} \pi (3r)^2 h = \frac{1}{3} \pi (9r^2) h = 3 \cdot 3 \cdot \left(\frac{1}{3} \pi r^2 h\right) = 9 \left(\frac{1}{3} \pi r^2 h\right)
\][/tex]
6. Factor Comparison:
Now, comparing [tex]\( V_2 \)[/tex] with [tex]\( V_1 \)[/tex], we can see that:
[tex]\[
V_2 = 9V_1
\][/tex]
Hence, the volume of the cone is multiplied by 9 when the radius is tripled and the height remains the same.
Therefore, the correct answer is:
[tex]\[
\text{A. 9}
\][/tex]
