Exponential Equations with like bases and by guess and check

Question

Substance                       Use                               Half-life
gallium-67               nuclear medicine                   80 hours

cobalt-60                 manufacturing                      5.3 years

technetium-99m       nuclear medicine                   6 hours

americium-241           construction                     432 years

carbon-14              archeological dating             5,715 years

uranium-235           atomic power                703,800,000 years


The formula for radioactive decay: A(t)=A₀e(in(.5)/T)t

where

A₀ is the amount initially present
T is the half-life of the substance
t is the time period over which the substance is studied
A(t) is the amount of the substance present after time tt


Using the table and formula from above solve the following:

Use guess and check to estimate how long will it take for 20% of a 1900-gram sample of americium-241 to decay?

It will take___________________ years

Answer 1

80% of 1900 = 1520

 

 

A(t) = 1900*(1/2)^(t/432) where t is in years

 

1520 = 1900 * (1/2)^(t/432)

 

1520/1900 = (1/2)^(t/432)

 

4/5 = (1/2)^(t/432)

 

log ( 4/5) = (t/432) * log(1/2)

 

log ( 4/5) / log(1/2) = t/432

 

t = 432*log(4/5)/log(1/2)

 

  = 139.073.....