The value of a particular investment follows a pattern of exponential growth. You invested money in a money market account. The value of your investment t years after your initial investment is given by the exponential growth model A=2600e^0.08t . By what percentage is the account increasing each year?
recall that a rate of r compounded continuously is e^rt
You are finding the equivalent annual rate equal to 8% per annum compounded continuously, that is
(1 + i)^t = e^.08t
take the tth root of both sides
1+i = e^.08 = 1.083287...
i = 0.832...
check, suppose we take 5 years
amount using your formula = 2600 e^5*.08) = 3878.74
amount using interest formul = 2600(1.083287..)^5 = 3878.74
well, well, ...
To determine the percentage by which the account is increasing each year, we need to find the annual growth rate expressed as a percentage.
The exponential growth model is given by A = 2600e^(0.08t), where A represents the value of the investment after t years.
The growth rate is equal to the coefficient of t in the exponent, which in this case is 0.08 (8% per year).
Therefore, the account is increasing by 8% each year.
To find the percentage by which the account is increasing each year, we can calculate the annual growth rate as a percentage of the initial investment.
The exponential growth model for the value of the investment is given by:
A = 2600e^(0.08t)
Here, "A" represents the value of the investment, and "t" represents the number of years since the initial investment.
To calculate the annual growth rate, we need to differentiate the equation with respect to time (t):
dA/dt = 2600 * 0.08 * e^(0.08t)
Now, let's substitute t = 0 to find the rate of growth at the initial investment:
dA/dt = 2600 * 0.08 * e^(0.08 * 0)
= 2600 * 0.08 * e^(0)
= 2600 * 0.08 * 1 (e^0 = 1, as any number raised to the power of 0 is 1)
= 208
So, at the initial investment, the rate of growth is 208 units per year.
To express the growth rate as a percentage, we can calculate the percentage increase by dividing the rate of growth by the initial investment (2600) and then multiplying by 100:
Percentage increase = (208 / 2600) * 100
= 8%
Therefore, the account is increasing by approximately 8% each year.