Question $6$: $(1$ point$)$

The half$-$life of a certain chemical substance on the planet Calculus is $55$ days. If a sample has an initial mass of $500500$ mg$,$ we want to determine the mass remaining after $1010$ days.

a$)$ The differential equation satisfied by the mass PP will depend on the relative growth $($or decay$)$ rate KK$.$ This differential equation is

dPdt$=$dPdt$=\_\_\_\_\_\_\_\_\_\_$

b$)$ If P$(0)=$CP$(0)=$C$,$ determine KK and write down the general solution of the differential equation.

P$($t$)=$P$($t$)=\_\_\_\_\_\_\_\_\_\_$

c$)$ If P$(0)=500$P$(0)=500,$ the mass remaining after $1010$ days is

P$(10)=$P$(10)=\_\_\_\_\_\_\_\_\_\_\_\_$

Write your answer with an accuracy of two decimals.