Calculus Question

Question

The half-life of cesium-137 is 30 years. Suppose we have a 150-mg sample.

(a) Find the mass that remains after t years.

y(t) =  ______ mg

(b) How much of the sample remains after 120 years? (Round your answer to two decimal places.)

_______mg

(c) After how long will only 1 mg remain? (Round your answer to one decimal place.)

t =  _______yr

Answer 1

1= 2e^30r

1/2= e^30r

ln.5 = 30r

r= ln.5/-30= -0.0231 =annual rate of decay

y= 150e^-.0231t

in 120 years y=150e^-0231(120) = 150(e^-2.772) = 150/16 = 75/8 =9.38 mg

120=4 half lives. in one half life 150 becomes 75

in 2 half lives, 75 becomes 37.5

in 3 half lives 37.5 becomes 18.75

in 4 half lives 18.75 becomes 9.375 mg

when does it become 1 mg?

in 5 half lives (150 years), 9.375 becomes 4.6875

in 6 half lives (180 years) 4.6875 becomes 2.34375

in 7 half lives (210 years) 2.34375 becomes 1.671875

in 8 half lives (240 years) 1.671875 becomes 08359375 mg

close to 240 years, maybe about 230 it becomes 1 mg

1 = 180e^-.0231t

ln(1/180) = -.0231t

t = ln(1/180)/-0231 =224.8 years to become 1 mg