Question 1157488: A family of 120 termites invades your house and grows at a rate of 18% per week. How many termites will be in your house after 1 year?
a) Use the rate formula to find the number of termites.
b) Use the rate to find the approximate Tdouble, and then find the number of termites in 1 year, using the Tdouble - new value formula.
c) Use the rate to find the exact Tdouble, and then find the number of termites in 1 year, using the Tdouble- new value formula.
d) Compare the values from a), b), and c).
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the formula to use is f = p * (1 + r) ^ n
f is the future value
p is the present value
r is the interest rate / growth rate per time period.
n is the number of time periods.
the time period is in weeks.
the growth rate per week is 18% = .18
in one year, the number of time periods is 52.
the formula becomes f = 120 * (1 + .18) ^ 52.
solve for f to get:
f = 656,214.203
to find out how long it takes for the number of termites to double, use the same formula and make f = 240, which is double 120.
the formula becomes 240 = 120 * (1 + .18) ^ n
divide both sides of this formula by 120 and simplify to get:
2 = (1 + .18) ^ n
take the log of both sides of this equation to get:
log(2) = log(1.18 ^ n)
by properties of logs, this becomes:
log(2) = n * log(1.18)
divide both sides of this equation by log(1.18) to get:
log(2) / log(1.18) = n
solve for n to get:
n = 4.187835134 weeks.
confirm by replacing n ih the origiinal equaion to get:
120 * 1.18 ^ 4.187835134 = 240.
the value of n is good.
if i understand the problem correctly, then the Tdouble formula is:
f = 120 * 1.18 ^ 4.187835134.
to find the number of termites after 1 year, then you need to divide 52 by 4.187835134 to get:
n = 12.4169167.
52 weeks is equivalent to 12.4169167 doublings of the termite population.
when the population doubles, the rate becomes 2, because 120 * 2 = 240.
the formula becomes f = 120 * 2 ^ 12.4169167.
solve for f to get:
f = 656,214.203
that's the same future value that we got, using the growth factor of 1.18 and the number of weeks as 52.
i'm not sure if this is what your looking for, but it does get you the same future value, only using the doubling growth rate rather than the weekly growth rate.
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