SOLUTION: The half- life of a radioactive substance is one hundred ninety-four days. How many days will it take for eighty percent of the substance to decay? I used the half life decay fo

The half- life of a radioactive substance is 
one hundred ninety-four days. How many days 
will it take for eighty percent of the substance 
to decay?  I used the half life decay formula.
my formula for the problem read...
0.8Nsubscript0=Nsubscript0*(1/2)superscript(t/194)
the result was about 62.5,
is this correct. 
=]

It couldn't be correct because it takes 194 days 
for 50% of the substance to decay. It would take 
much longer than 194 days for 80% of it to decay,
because in exponential decay, the rate of decay 
becomes slower and slower.

The general formula for any exponential growth
or decay is

A = Pert

Where P = beginning quantity in whatever quantity 
units are used,

A = final quantity in these units

r = a constant which is positive if the quantity
increases and negative if the quantity decreases.
We expect it to be negative since the quantity is
decreasing.

t = time in whatever time units are used.

Suppose we begin with P units of this substance.  
Then when t = 194 days, A = one half of P, or .5P  
So we substitute .5P for A and 194 for t:

.5P = Per(194)

Write r(194) as 194r and divide both sides by P

.5 = e194r

Use the rule that equation Y = eX can be written 
as equation X = ln(Y) to rewrite the above equation 
as

194r = ln(.5)

Solve for r

r = ln(.5)/194 

r = -.6931451806/194 =-.0035729236 units/day

So we replace this for r in the formula

A = Pert

A = Pe-.0035729236t

Now if we begin with P units of the substance, 
when 80% of it has decayed, the final amount 
will be 20% so we substitute .2P for A

.2P = Pe-.0035729236t

Divide both sides by P

.2 = e-.0035729236t 

Rewrite as

-.0035729236t = ln(.2)

Solve for t

t = ln(.2)/(-.0035729236)

t = 450.4540504 days

or about 450 days.

Edwin