The value of a particular investment follows a pattern of exponential growth. In the year 2000, you invested money in a money market account. The value of your investment t years after 2000 is given by the exponential growth model A=7200e^0.045t
A) How much money did you initially invest in the account?
B) How much would you have invested after 10, 20, and 30 years?
C) How long will it take for the invest to reach $10,200?
A) what is A(0)?
B) Plug in the values for t
C) solve for t in: 7200e^0.045t = 10200
anything^0 = 1 no matter what
so
at t = 0
A = 7200 * 1
I am not going to do them all for you
at t = 20 for example
A= A=7200e^0.045*20 = 7200 e^0.9 = 17,709.14
for part C
10,200 = 7200e^0.045t
102/72 = e^0.045t
ln (102/72) = 0.045 t
.3483/0.045 = t = 7.74 years
remember log base n of n^anything = the same anything
To answer these questions, we will use the exponential growth model A = P*e^(rt), where:
A = the value of the investment at a given time
P = the initial investment amount
r = the annual growth rate
t = the time in years
A) To find the initial investment amount:
We know that the value of the investment in the year 2000 (t = 0) is given as A = 7200.
Using the formula, we substitute t = 0 and A = 7200 to get:
7200 = P * e^(0.045 * 0)
To solve for P, we divide both sides by e^(0):
7200 / e^0 = P
P ≈ 7200 / 1 (since e^0 = 1)
P ≈ 7200
Therefore, the initial investment in the account was approximately $7,200.
B) To find the value of the investment after 10, 20, and 30 years:
We already have the equation A = 7200e^(0.045t).
For t = 10 years:
A = 7200e^(0.045 * 10)
Simply calculate this to find the value after 10 years.
For t = 20 years:
A = 7200e^(0.045 * 20)
Calculate this to find the value after 20 years.
For t = 30 years:
A = 7200e^(0.045 * 30)
Calculate this to find the value after 30 years.
C) To find how long it will take for the investment to reach $10,200:
We set A = 10,200 in the exponential growth model:
10,200 = 7200e^(0.045t)
To solve for t, we first divide both sides by 7200:
10,200 / 7200 = e^(0.045t)
Next, we take the natural logarithm (ln) of both sides:
ln(10,200 / 7200) = ln(e^(0.045t))
Using the property ln(e^x) = x, we simplify the equation:
ln(10,200 / 7200) = 0.045t
Finally, we divide both sides by 0.045 to isolate t:
t = ln(10,200 / 7200) / 0.045
Evaluate this expression using a calculator to find the approximate number of years it will take for the investment to reach $10,200.