Calculus Exponential Function Half-Life Question? Please help!?

Question

The half-life of Palladium-100 is 4 days. After 24 days a sample of Palladium-100 has been reduced to a mass of 2 mg. What is the mass 8 weeks after the start?

Can any math-wizard solve this problem and provide me a step-by-step solution please?

Answer 1

M(t) = Mo * (0.5)^(t/k)

Mo := M sub o or mass at t = 0

t := time passed

k := half-life of the object

M(Tn) = M(Tm) * (0.5)^(Kn-Km)/t)  

[read Tn := t sub n ; Tm := t sub m ; Kn := k sub n ; Km := k sub m]

8 weeks = 56 days

M(56) = M(24) * (0.5)^((56-24)/4)

M(56) = 2 * (0.5)^8

M(56) = 1/(2^7) = 0.0078125

Answer 2

T = P(1/2)^(t/h)

T = amount left

P = initial amount

t = time elapse

h = half life

T = P(1/2)^(t/4), where t is in days

given t = 24, and T = 2

2 = P(1/2)^(24/4)

P = 128

the intial amount is 128 mg

T = 128(1/2)^(t/4)

8 weeks = 56days

T = 128(1/2)^(56/4)

T = 0.0078125 mg ---  answer

hope it helps!

Answer 3

Since the half life is 4 days, 6 half lives have been competed since 0 hour. To get your 0 hour value, multiply 2 mg by 2^6 (because it is doubling 6 times). Now we must go back even further. There were 8 weeks (56 days), and in those weeks, 14 half lives were completed. So, from your 0 hour value, multiply by 2^14. This should be your final mass.

To sum it up, 20 half lives were completed so multiply 2 mg by 2^20.

=2097152 mg or 2097.152 grams or 2.097152 kg