Question 1143353: The half life of a substance is the time it takes a substance to decrease to half its initial amount. John has a pile of goo that decreases in amount at a constant rate. if John initially had 100 pounds of goo, and ten days later, he only had 25 pounds of goo, what Is the half life of the goo?
A)10
B) 5
C)7.5
D)20
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! if it decays at a constant rate, then you are dealing with an equation of the form f = p * (1 + r) ^ n.
f is the future value
p is the present value
r is the interest rate per time period
n is the number of time periods.
you start with:
f = 25
p = 100
n = 10
the formula becomes 25 = 100 * (1 + r) ^ 10
divide both sides of this formula by 100 to get:
.25 = (1 + r) ^ 10
take the 10th root of both sides of this equation to get:
.25 ^ (1/10) = 1 + r
subtract 1 from both sides of this equation to get:
.25 ^ (1/10) - 1 = r
solve for r to get:
r = -0.129449437
confirm by replacing r in the original equation with that to get:
25 = 100 * (1 - .129449437) ^ 10
this results in 25 = 25, confirming the solution is correct.
to find the half life, make f = .5 and p = 1 to get:
f = p * (1 + r) ^ n becomes .5 = (1 - .129449437) ^ n
take the log of both sides of the equation to get:
log(.5) = log((1 - .129449437) ^ n)
by the properties of logarithms, this becomes:
log(.5) = n * log(1 - .129449437)
solve for n to get:
n = log(.5) / log(1 - .129449437) = 5
your solution is that the half life of the goo is 5 days.
the goo deteriorates at the constant rate of 12.9449437% per day.
that means it loses 12.9449437% of its mass every day.
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