SOLUTION: A radioactive substance is produced from nuclear fallout. If 250g of this substance decays to 150g in 30years, what is the half-life of this substance? Solve algebraically usinglog

Question 1206895: A radioactive substance is produced from nuclear fallout. If 250g of this substance decays to 150g in 30years, what is the half-life of this substance? Solve algebraically usinglogarithms. Answer to 2 decimal places.
Found 3 solutions by josgarithmetic, greenestamps, math_tutor2020:
Answer by josgarithmetic(39621)   (Show Source): You can put this solution on YOUR website!

If,...

Half-Life?

---------To 2 decimal places? 40.71 years
Answer by greenestamps(13203)   (Show Source): You can put this solution on YOUR website!

Let x be the number half-lives in 30 years. Then, since 250g decays to 150g in 30 years,

= 0.7369656 to several decimal places

The half life is then

= 40.71 to 2 decimal places

ANSWER: 40.71 years

Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

Answer: 40.71 years

Explanation

One way to represent the half-life formula is:
y = A*(0.5)^(x/H)
where,
x = number of years
A = starting amount
H = half-life in years
y = final amount after x years
Optionally you can replace "years" with some other time unit.

Let's isolate H.
y = A*(0.5)^(x/H)
y/A = (0.5)^(x/H)
log( y/A ) = log( (0.5)^(x/H) )
log( y/A ) = (x/H)*log( 0.5 )
H*log( y/A ) = x*log( 0.5 )
H = x*log( 0.5 )/log( y/A )

Plug in x = 30, y = 150, A = 250
You should get the following
H = x*log( 0.5 )/log( y/A )
H = 30*log( 0.5 )/log( 150/250 )
H = 30*log( 0.5 )/log( 0.6 )
H = 40.707463465702 approximately
H = 40.71 years is the approximate half-life when rounding to 2 decimal places.

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