Answer
(a) $N = 9.17\times 10^9$
(b) $N = 4.20\times 10^9$
(c) $N = 1.73\times 10^6$
Work Step by Step
(a) We can find the number of $^{226}Ra$ atoms remaining after $t = 200~years$:
$N = N_0~(0.5)^{t/(t_{1/2})}$
$N = (1.00\times 10^{10})~(0.5)^{200/1600}$
$N = 9.17\times 10^9$
(b) We can find the number of $^{226}Ra$ atoms remaining after $t = 2000~years$:
$N = N_0~(0.5)^{t/(t_{1/2})}$
$N = (1.00\times 10^{10})~(0.5)^{2000/1600}$
$N = 4.20\times 10^9$
(c) We can find the number of $^{226}Ra$ atoms remaining after $t = 20,000~years$:
$N = N_0~(0.5)^{t/(t_{1/2})}$
$N = (1.00\times 10^{10})~(0.5)^{20,000/1600}$
$N = 1.73\times 10^6$
