The Concord Community Park Association receives a $10,000 donation from an anonymous donor who stipulates that before using the funds to improve the park, they must invest the money until it has doubled in value. The Park Association decides to invest the money in an account that pays 7%, compounded quarterly.
a. Write an equation that models the growth in value of the donation over time.
b. How many years will it take for the initial donation to double?
a. P = Po(1+r)^n.
b. Po = $10,000 = Initial deposit.
r = (7%/4)/100% = 0.0175 = Quarterly %
rate expressed as a decimal.
n = The # of compounding periods.
P = 10000*(1.0175)^n = $20,000.
(1.0175)^n = 2.0
n*Log(1.0175) = Log2
n = Log2/Log(1.0175)=40 Comp. periods.
T = 40Comp. / 4comp./yr = 10 Years.
a. To model the growth in value of the donation over time, we can use the following formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the total amount after time t
P = the principal amount (initial donation)
r = the annual interest rate (7% = 0.07)
n = the number of times interest is compounded per year (quarterly = 4 times)
t = the number of years
Using this formula, the equation that models the growth in value of the donation over time is:
A = 10000(1 + 0.07/4)^(4t)
b. We want to find the number of years (t) it will take for the initial donation to double, which means the final amount (A) should be twice the initial donation (2P).
Setting up the equation:
2P = P(1 + 0.07/4)^(4t)
Dividing both sides by P, we get:
2 = (1 + 0.07/4)^(4t)
Taking the natural logarithm (ln) of both sides to solve for t:
ln(2) = ln[(1 + 0.07/4)^(4t)]
Using the logarithmic property, we can bring down the exponent:
ln(2) = (4t) * ln(1 + 0.07/4)
Dividing both sides by 4 * ln(1 + 0.07/4), we can isolate t:
t = [ln(2)] / [4 * ln(1 + 0.07/4)]
Plugging the values into a calculator, we can find the value of t.