Both uranium$-235$ and plutonium$-239$ can undergo fission and are used to make atomic bombs. The uranium$-235$ is obtained from natural uranium ore while plutonium is a byproduct of some reactors such as the Canadian Candu reactors. Reactors that produce plutonium that can be used as weapons are called breeder reactors. Plutonium$-239$ has a half$-$life of $24110$ years and undergoes alpha and gamma decay.

$3.$ The first nuclear weapon was detonated on July $16,1945$ at Trinity test site in Alamogordo, New Mexico. It used $6\mathrm{.}2$ kg of plutonium$-239$ with an explosive power equivalent to $20$ kilotons of TNT$.$ 

a$.$ Write the decay equation for plutonium$-239.$

b$.$ Determine the amount of plutonium$-239$ remaining after $10000$ years, starting with $6\mathrm{.}2$ kg$.$