To determine the factor by which the dimensions of cylinder [tex]\(A\)[/tex] are multiplied to produce the corresponding dimensions of cylinder [tex]\(B\)[/tex], we need to understand and find the radii of the bases of both cylinders.
1. Find the radius of cylinder [tex]\(A\)[/tex]:
- We know the circumference of the base of cylinder [tex]\(A\)[/tex] is [tex]\(4\pi\)[/tex] units.
- The formula for the circumference [tex]\(C\)[/tex] of a circle is given by:
[tex]\[
C = 2\pi r
\][/tex]
- Solving for the radius [tex]\(r\)[/tex]:
[tex]\[
4\pi = 2\pi r \implies r = \frac{4\pi}{2\pi} = 2 \text{ units}
\][/tex]
Thus, the radius of the base of cylinder [tex]\(A\)[/tex] is [tex]\(2\)[/tex] units.
2. Find the radius of cylinder [tex]\(B\)[/tex]:
- We know the area of the base of cylinder [tex]\(B\)[/tex] is [tex]\(9\pi\)[/tex] square units.
- The formula for the area [tex]\(A\)[/tex] of a circle is given by:
[tex]\[
A = \pi r^2
\][/tex]
- Solving for the radius [tex]\(r\)[/tex]:
[tex]\[
9\pi = \pi r^2 \implies r^2 = 9 \implies r = \sqrt{9} = 3 \text{ units}
\][/tex]
Thus, the radius of the base of cylinder [tex]\(B\)[/tex] is [tex]\(3\)[/tex] units.
3. Calculate the multiplication factor:
- To determine the factor by which the dimensions of cylinder [tex]\(A\)[/tex] are multiplied to produce cylinder [tex]\(B\)[/tex], we take the ratio of the radii of the two cylinders.
[tex]\[
\text{Factor} = \frac{\text{Radius of } B}{\text{Radius of } A} = \frac{3}{2}
\][/tex]
Therefore, the dimensions of cylinder [tex]\(A\)[/tex] are multiplied by the factor [tex]\(\frac{3}{2}\)[/tex] to produce the dimensions of cylinder [tex]\(B\)[/tex].
So, the answer is [tex]\(\boxed{\frac{3}{2}}\)[/tex].
