The half-life of a certain radioactive material is 37 days. An initial amount of the material has a mass of 477 kg. Write an exponential function that models the decay of this material. Find how much radioactive material remains after 6 days. Round your answer to the nearest thousandth.

Question

campbell_st · Answer

ok... so the model will be 

$$P = P_{0} e^{-kt} $$

so 

P is population, Po is initial population, k is the growth/decay constant and t = time

you need to find the value of k... using the half life information. 

so t = 37, Po = 477 and P = 238.5  (half life quantity


$$238.5 = 477e^{-37k}$$

so 

$$0.5 = e^{-37t}$$

can you solve for t..?

anonymous · Answer

im going to be honest i have no idea what to do..

campbell_st · Answer

oops slight typo above ... we need to solve for k... the decay constant

the model should read 

$$0.5 = e^{-37k}$$

ok... to solve for k you need to take the base e log of both sides

this will remove e

so 

ln(0.5) = 37k

so k = -ln(0.5)/37

so k = 0.018734

so you exponential model is 

$$P = 447e^{-0.018734t}$$

all you need to do is substitute t = 6 and evaluate

hope this helps

campbell_st · Answer

to make it easy the model is

$$P = 447 \times e^{-0.018734 \times t}$$

anonymous · Answer

0 kg
426.288 kg
0.736 kg
0.184 kg

my answer didnt match any of these i got 399.476?

campbell_st · Answer

the way to get the correct answer is to use brackets... 

$$P = 477\times e^{(-0.018734 \times 6)}$$