Question

2 practice 2
an oblique cylinder with a base of radius 2 units is shown.
the top of the cylinder can be obtained by translating the base by the directed line segment ab which has length $6\sqrt{2}$ units. the segment ab forms a $45^{\circ}$ angle with the plane of the base. what is the volume of the cylinder? give the exact answer.
type your answer in the box.
_____ cubic units

Explanation:

Step1: Find height of the cylinder

The height $h$ is the component of $AB$ perpendicular to the base. Using $\sin(45^\circ)=\frac{h}{AB}$, substitute $AB=6\sqrt{2}$ and $\sin(45^\circ)=\frac{\sqrt{2}}{2}$:
$$h = 6\sqrt{2} \times \frac{\sqrt{2}}{2}$$
$$h = 6\sqrt{2} \times \frac{\sqrt{2}}{2} = 6 \times \frac{2}{2} = 6$$

Step2: Calculate base area

Base is a circle with radius $r=2$. Area formula: $A=\pi r^2$
$$A = \pi \times 2^2 = 4\pi$$

Step3: Compute cylinder volume

Volume formula for oblique cylinder: $V = A \times h$
$$V = 4\pi \times 6 = 24\pi$$

Answer:

$24\pi$ cubic units