2021 Fermat Problem 22

An unpainted cone has radius 3 cm and slant height 5 cm. The cone is placed in a container of paint. With the cone's circular base resting flat on the bottom of the container, the depth of the paint in the container is 2 cm. When the cone is removed, its circular base and the lower portion of its lateral surface are covered in paint. The fraction of the total surface area of the cone that is covered in paint can be written as \dfrac{p}{q} where p and q are positive integers with no common divisor larger than 1. What is the value of p+q?

(The *lateral surface* of a cone is its external surface not including the circular base. A cone with radius r, height h, and slant height s has lateral surface area equal to \pi r s.)

\textbf{(A)}\ 59\quad \textbf{(B)}\ 61\quad \textbf{(C)}\ 63\quad \textbf{(D)}\ 65\quad \textbf{(E)}\ 67

If there are no answer choices shown, enter a numerical answer.

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