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A wedge is cut out of a circular cylinder of radius 10 by two planes. one plane is perpendicular to the axis of the cylinder. the other intersects the first at an angle of 30° along a diameter of the cylinder. find the volume v of the wedge, taking rectangular cross-sections to be parallel to the line of intersection of the two planes, which is the x-axis. (let r = 10.)

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nobillionaireNobley
The radius of the circular cylinder is given by:

[tex]r^2=x^2+y^2 \\ \\ \Rightarrow y=\sqrt{10^2-x^2}=\sqrt{100-x^2}[/tex]

The wedge makes a shape whose cross-section perpendicular to the x -axis at distance x from the origin is a triangle whose base is is given by: [tex]y=\sqrt{100-x^2}[/tex]

The height of the wedge is given by:

[tex]y\tan30^o= \frac{\sqrt{100-x^2}}{\sqrt{3}} [/tex]

The cross sectional area of the wedge is given by:

[tex]A(x)= \frac{1}{2} bh \\ \\ = \frac{\sqrt{100-x^2}}{2} \cdot\frac{\sqrt{100-x^2}}{\sqrt{3}} \\ \\ = \frac{100-x^2}{2\sqrt{3}} [/tex]

The volume is given by:

[tex]V= \int\limits^{10}_{-10} {A(x)} \, dx \\ \\ = \int\limits^{10}_{-10} {\frac{100-x^2}{2\sqrt{3}}} \, dx =2\int\limits^{10}_{0} {\frac{100-x^2}{2\sqrt{3}}} \, dx \\ \\ =\int\limits^{10}_{0} {\frac{100-x^2}{\sqrt{3}}} \, dx=\frac{1}{\sqrt{3}}\left[100x-\frac{x^3}{3}\right]^{10}_0 \\ \\ =\frac{1}{\sqrt{3}}\left[100(10)-\frac{(10)^3}{3}\right]=\frac{1}{\sqrt{3}}\left[1000-\frac{1000}{3}\right] \\ \\ =\frac{1}{\sqrt{3}}\left[\frac{2000}{3}\right]=\frac{2000}{3\sqrt{3}}[/tex]

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