Graphs of Functions and Optimization Problems

(cod: P-61-46-6) A conical-shaped cup is to be constructed as follows: from a circular sheet with radius \(R = 9 \, cm\), a circular sector \(OAB\) with central angle \(\theta\) is cut out. The remaining part of the sheet is converted into a cone by aligning \(OA\) with \(OB\), as shown in the figure below.

Determine the value of \(\theta\) in radians that maximizes the volume of the cup.

\(\theta = 2\pi \left(1- \sqrt{\dfrac{4}{5}}\right).\)

\(\theta = \pi \left(1- \sqrt{\dfrac{2}{3}}\right).\)

\(\theta = 2\pi \left(1- \sqrt{\dfrac{3}{5}}\right).\)

\(\theta = \pi \left(1- \sqrt{\dfrac{3}{5}}\right).\)

\(\theta = 2\pi \left(1- \sqrt{\dfrac{2}{3}}\right).\)