Question 1103994: A population increases from 21,000 to 27,000 over a 5-year period at a constant annual percent growth rate.
a) By what percent did the population increase in total?
5.155% (round to the nearest 0.001%)
b) At what constant percent rate of growth did the population increase each year?
5.155% (round to the nearest 0.001%)
c) At what continuous annual growth rate did this population grow?
0.050% (round to the nearest 0.001%)
I included the answers I got, but I know that they are not all right.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! A population increases from 21,000 to 27,000 over a 5-year period at a constant annual percent growth rate.
a) By what percent did the population increase in total?
27,000 / 21,000 = 1.285714286 - 1 = .285714286 * 100 = 28.5714286% rounded to 3 decimal digits = 28.571%.
they were not asking for the annual rate of increase.
they just wanted the total rate of increase.
b) At what constant percent rate of growth did the population increase each year?
27,000 / 21,000 = 1.285714286 ^ (1/5) = 1.051547497 - 1 = .051547497 * 100 = 5.1547497% rounded to 3 decimal digits = 5.155%.
you got this one right.
you could also have solved it this way.
the discrete compounding formula is f = p * (1 + r) ^ n
r is the future value
p is the present value
r is the interest rate per time period
n is the number of time periods.
your time periods are in years, therefore no adjustment is made to r or n.
your formula becomes:
27,000 = 21,000 * (1 + r) ^ 5
divide both sides of this equation by 21,000 to get:
27,000 / 21,000 = (1 + r) ^ 5
take the fifth root of both sides of this equation to get:
(27,000 / 21,000) ^ (1/5) = 1 + r
subtract 1 from both sides of this equation to get:
(27,000 / 21,000) ^ (1/5) - 1 = r
solve for r to get:
r = .0515474968 * 100 = 5.15474968% rounded to 2 decimal places = 5.155%.
c) At what continuous annual growth rate did this population grow?
the continuous growth rate formula is f = p * e^(r*n)
p is the present value.
f is the future value.
e is the scientific constant of 2.718281828.......
r is the interest rate per time period
n is the number of time periods.
your present value is 21,000 and your future value is 27,000 and your number of years is 5.
the formula becomes 27,000 = 21,000 * e^(r * 5)
divide both sides of this equation by 21,000 to get:
27,000 / 21,000 = e^(r * 5)
take the natural log of both sides of this equation to get:
ln(27,000 / 21,000) = ln(e^(5r))
since ln(e^x) = x*ln(e), your equation becomes:
ln(27,000 / 21,000) = 5r * ln(e)
since ln(e) = 1, your equation becomes:
ln(27,000 / 21,000) = 5r
divide both sides of this equation by 5 to get:
ln(27,000 / 21,000) / 5 = r
solve for r to get:
r = .0502628857 * 100 = 5.026728857% rounded to 3 decimal places = 5.027%.
the more number of compounding periods per year you have, the higher the effective interest rate.
that is why you get the same future value with the interest rate of 5.155% compounded annually as you get with the interest rate of 5.027% compounded continuously.
compounding continuously is the most compounding periods per year you can get.
if you did the discrete compounding formula with monthly compounding, you would have gotten an interest rate between 5.155% and 5.027%.
with monthly compounding, you formula would have become:
27,000 = 21,000 * (1 + r/12) ^ (5*12)
that would have become:
27,000 / 21,000 = (1 + r/12) ^ 60)
you would have taken the 60th root of both sides of this equation to get:
(27,000 / 21,000) ^ (1/60) = 1 + r/12
subtract 1 from both sides of this equation to get:
(27,000 / 21,000) ^ (1/60) - 1 = r/12.
solve for r/12 to get:
r/12 = .0041973581
solve for r to get:
r = .0503682977 * 100 = 5.03682977% rounded to 3 decimal digits = 5.037%.
with annual compounding, your interest rate had to be 5.155%.
with monthly compounding, your interest rate had to be 5.037%.
with continuous compounding, your interest rate had to be 5.027%.
the formula for discrete compounding is:
f = p * (1 + r) ^ n
r is the interest rate per time period.
n is the number of time periods.
you divide the annual rate by the number of compounding periods per year to get the interest rate per time period.
you multiply the number of years by the number of compounding periods per year to get the number of time periods.
this formula requires the time period adjustments be made before using the formula.
the formula can also be shown as:
f = p * (1 + r/c) ^ (n*c)
in this version of the formula, r is the annual interest rate and n is the number of years and c is the number of compounding periods per year.
this version of the formula assumes annual interest rate and number of year and adjusts for the number of compounding periods per year as part of the formula.
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