question_answer

The lateral surface of a cylinder develops into a rectangle whose diagonal is equal to 4x and forms an angle \[\frac{x}{6}\] with the base. What is the volume of the cylinder?

A)  \[6{{x}^{2}}\]                       

B)  \[18{{x}^{2}}\]

C)  \[36{{x}^{2}}\]                     

D)  \[72{{x}^{2}}\]

Correct Answer: A

Solution :

Let r and h be the radius and height of the cylinder in \[\Delta ABC.\]             \[\sin 30{}^\circ =\frac{BC}{AC}\]             \[\gamma =2\pi \] And      \[\cos 30{}^\circ =\frac{AB}{AC}\] Or         \[\frac{\sqrt{3}}{2}=\frac{x}{4\pi }\] Or         \[x=2\sqrt{3}\pi \] \[\therefore \]      \[2\pi r=2\sqrt{3}\pi \] Or         \[r=\sqrt{3}\] and \[h=2\pi \] \[\therefore \] Volume of cylinder \[=\pi {{r}^{2}}h\]                         \[=\pi \times {{(\sqrt{3})}^{2}}\times 2\pi \]                         \[=6{{\pi }^{2}}\]