Answer:
Step-by-step explanation:
To find the volume of the cylinder, we need to know the radius and height of the cylinder. Given that the directed line segment AB forms a 45° angle with the plane of the base and has a length of \(6\sqrt{2}\), we can determine the radius and height of the cylinder.
Let's denote the radius of the cylinder as \(r\) and the height of the cylinder as \(h\).
Since segment AB forms a 45° angle with the plane of the base, it creates a right triangle with the base of the cylinder. The length of the hypotenuse of this right triangle is \(6\sqrt{2}\), which represents the slant height of the cylinder.
We can use trigonometric ratios to find the radius and height of the cylinder:
\[ \text{cos}(45^\circ) = \frac{\text{adjacent side}}{\text{hypotenuse}} \]
\[ \text{cos}(45^\circ) = \frac{r}{6\sqrt{2}} \]
\[ r = 6 \]
So, the radius of the cylinder is \(6\).
Now, we can use the Pythagorean theorem to find the height of the cylinder:
\[ \text{height} = \sqrt{(\text{hypotenuse})^2 - (\text{base})^2} \]
\[ h = \sqrt{(6\sqrt{2})^2 - 6^2} \]
\[ h = \sqrt{72 - 36} \]
\[ h = \sqrt{36} \]
\[ h = 6 \]
So, the height of the cylinder is \(6\).
Now that we have the radius and height of the cylinder, we can calculate its volume using the formula:
\[ \text{Volume} = \pi r^2 h \]
\[ \text{Volume} = \pi (6)^2 (6) \]
\[ \text{Volume} = 36\pi \times 6 \]
\[ \text{Volume} = 216\pi \]
So, the volume of the cylinder is \(216\pi\).