To determine how the volume of an oblique cylinder changes when the radius is reduced to [tex]\(\frac{2}{9}\)[/tex] of its original size and the height is quadrupled, we need to understand the formula for the volume of a cylinder.
The volume [tex]\(V\)[/tex] of a cylinder is given by:
[tex]\[ V = \pi r^2 h \][/tex]
where:
- [tex]\(r\)[/tex] is the radius of the cylinder's base,
- [tex]\(h\)[/tex] is the height of the cylinder.
Let's consider the initial volume with radius [tex]\(r\)[/tex] and height [tex]\(h\)[/tex]:
[tex]\[ V_{\text{initial}} = \pi r^2 h \][/tex]
When the radius is changed to [tex]\(\frac{2}{9}\)[/tex] of its original size:
[tex]\[ \text{New radius} = \frac{2}{9} r \][/tex]
And the height is quadrupled:
[tex]\[ \text{New height} = 4h \][/tex]
Now, we find the new volume with these changes. The new volume [tex]\(V_{\text{new}}\)[/tex] is given by:
[tex]\[ V_{\text{new}} = \pi \left(\frac{2}{9} r\right)^2 \left(4h\right) \][/tex]
First, calculate [tex]\(\left(\frac{2}{9} r\right)^2\)[/tex]:
[tex]\[ \left(\frac{2}{9} r\right)^2 = \left(\frac{2}{9}\right)^2 r^2 = \frac{4}{81} r^2 \][/tex]
Next, multiply by the new height:
[tex]\[ V_{\text{new}} = \pi \left(\frac{4}{81} r^2\right) (4h) \][/tex]
[tex]\[ V_{\text{new}} = \pi \left(\frac{4 \times 4}{81}\right) r^2 h \][/tex]
[tex]\[ V_{\text{new}} = \pi \left(\frac{16}{81}\right) r^2 h \][/tex]
[tex]\[ V_{\text{new}} = \frac{16}{81} \pi r^2 h \][/tex]
Hence, the volume of the oblique cylinder changes to:
[tex]\[ V_{\text{new}} = \frac{16}{81} \pi r^2 h \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{B. \ \ V = \frac{16}{81} \pi r^2 h} \][/tex]
