Sure! Let's solve this step-by-step.
First, let's recall the formula for the volume of a cylinder:
[tex]\[ V = \pi r^2 h \][/tex]
where [tex]\( r \)[/tex] is the radius and [tex]\( h \)[/tex] is the height of the cylinder.
Now given:
1. The radius is quadrupled, which means the new radius [tex]\( r' \)[/tex] is [tex]\( 4r \)[/tex].
2. The height is reduced to a third of its original size, so the new height [tex]\( h' \)[/tex] is [tex]\( \frac{h}{3} \)[/tex].
We need to find the new volume [tex]\( V' \)[/tex] using these new dimensions. Let's plug these values into the volume formula:
[tex]\[ V' = \pi (r')^2 h' \][/tex]
[tex]\[ V' = \pi (4r)^2 \left(\frac{h}{3}\right) \][/tex]
Next, simplify inside the parentheses:
[tex]\[ V' = \pi (16r^2) \left(\frac{h}{3}\right) \][/tex]
Multiply the terms:
[tex]\[ V' = \pi \cdot 16r^2 \cdot \frac{h}{3} \][/tex]
[tex]\[ V' = \frac{16}{3} \pi r^2 h \][/tex]
Thus, the new volume [tex]\( V' \)[/tex] is [tex]\( \frac{16}{3} \)[/tex] times the original volume [tex]\( V \)[/tex]:
[tex]\[ V' = \frac{16}{3} V \][/tex]
[tex]\[ V' = \frac{16}{3} \pi r^2 h \][/tex]
Therefore, the correct answer to the question is:
D. [tex]\( V = \frac{16}{3} \pi r^2 h \)[/tex]
