Problem#$3$: $(5$ Marks$)$

Archimedes supposedly was asked to determine whether a crown made for the king consisted of pure gold. According to legend, he solved this problem by weighing the crown first in air and then in water as shown in the figure. Suppose the scale read $7\mathrm{.}84$ N when the crown was in air and $6\mathrm{.}84$ N when it was in water.

a$)$ Determine the buoyant force acting on the crown? $(1$ Marks$)$

b$)$ Find the density of the crown? $(4$ Marks$)$