math precalculus 12

Question

The half-life of a radio active substance is 5 weeks. Originally there is 175g of this substance. How long before only 30g remains?

Answer 1

We will use the equation for the decay of radioactive materials:

N = N_o e^-λt

N - the amount of radioactive material left

N_o - the initial amount of radioactive material at time zero

λ - decay coefficient

t - time

λ = 0.693/T_1/2

T_1/2 - half time of radioactive material

Calculating λ:

There will be 35 days in 5 weeks.

λ = 0.693/35 = 0.0198

Writing the equation:

30 = 175 • e^-0.0198t → divide by "175"

(30/175) = e^-0.0198t → take ln

ln(30/175) = ln(e^-0.0198t)

-1.76358859 = -0.0198t

t = 89.07 (days) or approximately 12.72 (weeks)

Answer 2

30 = 175(0.5)^t/5

0.171428571 ≈ 0.5^t/5

ln 0.171428571 ≈ (t/5)(ln 0.5)

Can you solve for t and answer?