Let's start by reviewing the volume formula for a cylinder. The volume [tex]\( V \)[/tex] of a cylinder with radius [tex]\( r \)[/tex] and height [tex]\( h \)[/tex] is given by:
[tex]\[ V = \pi r^2 h \][/tex]
Now, let's analyze the changes:
1. The radius [tex]\( r \)[/tex] is reduced to [tex]\(\frac{2}{5}\)[/tex] of its original size.
So, the new radius [tex]\( r' \)[/tex] is:
[tex]\[ r' = \frac{2}{5} r \][/tex]
2. The height [tex]\( h \)[/tex] is quadrupled.
So, the new height [tex]\( h' \)[/tex] is:
[tex]\[ h' = 4h \][/tex]
We now want to determine the new volume [tex]\( V' \)[/tex] of the cylinder with the new dimensions. Using the new radius [tex]\( r' \)[/tex] and new height [tex]\( h' \)[/tex], the volume [tex]\( V' \)[/tex] is:
[tex]\[ V' = \pi (r')^2 h' \][/tex]
Substitute [tex]\( r' = \frac{2}{5} r \)[/tex] and [tex]\( h' = 4h \)[/tex]:
[tex]\[ V' = \pi \left( \frac{2}{5} r \right)^2 (4h) \][/tex]
First, square the new radius:
[tex]\[ \left( \frac{2}{5} r \right)^2 = \frac{4}{25} r^2 \][/tex]
Now, substitute this back into the volume formula:
[tex]\[ V' = \pi \left( \frac{4}{25} r^2 \right) (4h) \][/tex]
Distribute the [tex]\( 4h \)[/tex]:
[tex]\[ V' = \pi \left( \frac{4}{25} \right) \left( 4 r^2 h \right) \][/tex]
Simplify inside the parentheses:
[tex]\[ V' = \pi \left( \frac{16}{25} r^2 h \right) \][/tex]
Rewriting it:
[tex]\[ V' = \frac{16}{25} \pi r^2 h \][/tex]
Recall that [tex]\( \pi r^2 h \)[/tex] is just the original volume [tex]\( V \)[/tex]:
[tex]\[ V' = \frac{16}{25} V \][/tex]
Thus, the new volume of the cylinder is [tex]\(\frac{16}{25}\)[/tex] of the original volume. Therefore, the correct answer is:
B. The volume is now [tex]\(\frac{16}{25}\)[/tex] the original volume.
