Consider rhombus ABCD with side length 6:

This is rolled to form a cylinder of volume \(\frac{27}{2}\) by taping \(\overline{AD}\) to \(\overline{BC}\):

What is \(\sin\left(\angle BAD \right)\)?

 Aug 19, 2023

The cylinder formed has height h and radius r. Since the volume of the cylinder is 27/2​, we have:

\pi r^2 h = \frac{27}{2}

Since ABCD is a rhombus, we know that AB=BC=6 and AC=BD=2*sqrt(6)​. The circumference of the cylinder's base is 2πr=AB+BC=12, so r=6/π.

Substituting this into the equation for the volume of the cylinder, we get:

\pi \left( \frac{6}{\pi} \right)^2 h = \frac{27}{2}

h = \frac{27/2}{9/\pi} = \frac{2\pi}{9}

The angle BAD is the angle between the bases of the cylinder. Since the height of the cylinder is h=2π/9​, the sine of angle BAD is:

\sin \angle BAD = \frac{h}{2r} = \frac{2\pi/9}{6/\pi} = \boxed{\frac{\pi}{9}}

 Aug 19, 2023

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