To determine the scaling factor that transforms the dimensions of cylinder [tex]\(A\)[/tex] into the dimensions of cylinder [tex]\(B\)[/tex], we need to follow a series of steps.
1. Compute the radius of cylinder [tex]\(A\)[/tex]:
The circumference of cylinder [tex]\(A\)[/tex] is given as [tex]\(4 \pi\)[/tex] units. The formula for the circumference of a circle is:
[tex]\[
\text{Circumference} = 2 \pi r
\][/tex]
Where [tex]\(r\)[/tex] is the radius. We can solve for [tex]\(r\)[/tex] by setting:
[tex]\[
2 \pi r = 4 \pi
\][/tex]
Dividing both sides by [tex]\(2 \pi\)[/tex]:
[tex]\[
r = \frac{4 \pi}{2 \pi} = 2
\][/tex]
Thus, the radius of the base of cylinder [tex]\(A\)[/tex] is [tex]\(2\)[/tex] units.
2. Compute the radius of cylinder [tex]\(B\)[/tex]:
The area of the base of cylinder [tex]\(B\)[/tex] is given as [tex]\(9 \pi\)[/tex] units. The formula for the area of a circle is:
[tex]\[
\text{Area} = \pi r^2
\][/tex]
Where [tex]\(r\)[/tex] is the radius. We can solve for [tex]\(r\)[/tex] by setting:
[tex]\[
\pi r^2 = 9 \pi
\][/tex]
Dividing both sides by [tex]\(\pi\)[/tex]:
[tex]\[
r^2 = 9
\][/tex]
Taking the square root of both sides:
[tex]\[
r = \sqrt{9} = 3
\][/tex]
Thus, the radius of the base of cylinder [tex]\(B\)[/tex] is [tex]\(3\)[/tex] units.
3. Determine the scaling factor:
The scaling factor is the ratio of the radius of cylinder [tex]\(B\)[/tex] to the radius of cylinder [tex]\(A\)[/tex]:
[tex]\[
\text{Scaling factor} = \frac{\text{radius of cylinder } B}{\text{radius of cylinder } A} = \frac{3}{2}
\][/tex]
Thus, the factor by which the dimensions of cylinder [tex]\(A\)[/tex] are scaled to produce the corresponding dimensions of cylinder [tex]\(B\)[/tex] is [tex]\(\boxed{\frac{3}{2}}\)[/tex].
