Question

1. 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 ( 6sqrt{2} ) units. the segment ab forms a ( 45^{circ} ) angle with the plane of the base. what is the volume of the cylinder?

Explanation:

Step1: Find the height of the cylinder

We know that for a line - segment \(AB\) which makes an angle \(\theta\) with the plane of the base of the cylinder, the height \(h\) of the cylinder (perpendicular distance between the two bases) can be found using the formula \(h = |AB|\sin\theta\).
Given \(|AB|=6\sqrt{2}\) units and \(\theta = 45^{\circ}\), and \(\sin45^{\circ}=\frac{\sqrt{2}}{2}\).
So, \(h = 6\sqrt{2}\times\frac{\sqrt{2}}{2}\).

$$h=\frac{6\times2}{2}=6$$

Step2: Find the area of the base

The area of the base of a cylinder (a circle) is given by the formula \(A=\pi r^{2}\). Given \(r = 2\) units.
So, \(A=\pi\times(2)^{2}=4\pi\)

Step3: Calculate the volume of the cylinder

The volume of a cylinder (both right - circular and oblique) is given by the formula \(V=Ah\), where \(A\) is the area of the base and \(h\) is the height.
Substituting \(A = 4\pi\) and \(h = 6\) into the formula, we get \(V=4\pi\times6\)

$$V = 24\pi$$

Answer:

\(24\pi\)