To solve this question, we'll use the formula for the area of a circle and the formula for the volume of a cylinder. Let's go through the problem step-by-step.
Firstly, we're given that the area of the base of the cylinder is [tex]\(121\pi \text{ cm}^2\)[/tex]. The formula for the area of a circle is:
[tex]\[ A = \pi r^2 \][/tex]
where [tex]\( A \)[/tex] is the area and [tex]\( r \)[/tex] is the radius of the circle.
Knowing the area, we want to find the radius. By comparing the given area of the base with the formula for the area of a circle, we can write:
[tex]\[ 121\pi = \pi r^2 \][/tex]
We can divide both sides of the equation by [tex]\( \pi \)[/tex] to solve for [tex]\( r^2 \)[/tex]:
[tex]\[ 121 = r^2 \][/tex]
Then, taking the square root of both sides, we find the radius:
[tex]\[ r = \sqrt{121} \][/tex]
[tex]\[ r = 11 \text{ cm} \][/tex]
Now we are given that the height of the cylinder ([tex]\( h \)[/tex]) is four times the radius:
[tex]\[ h = 4r \][/tex]
[tex]\[ h = 4 \cdot 11 \text{ cm} \][/tex]
[tex]\[ h = 44 \text{ cm} \][/tex]
Finally, we can now calculate the volume of the cylinder ([tex]\( V \)[/tex]) using the formula:
[tex]\[ V = \pi r^2 h \][/tex]
Plugging in the values we have for [tex]\( r \)[/tex] and [tex]\( h \)[/tex]:
[tex]\[ V = \pi \cdot 11^2 \cdot 44 \][/tex]
[tex]\[ V = \pi \cdot 121 \cdot 44 \][/tex]
[tex]\[ V = 5324\pi \text{ cm}^3 \][/tex]
Therefore, the volume of the cylinder is [tex]\( 5324\pi \text{ cm}^3 \)[/tex]. This is the exact answer, using [tex]\( \pi \)[/tex] in the result.
