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 = 1600e^(0.052t). By what percentage is the account increasing each year?
By the "just look at it" theorem, I would say
5.2% as an instantaneous rate.
If you want the rate per annum, compounded annually .....
e^(.052(1)) = 1.0533775...
so appr 5.34%
To find the percentage increase of the account each year, we need to determine the rate of change of the investment value with respect to time. In this case, we have the exponential growth model for the value of the investment:
A = 1600e^(0.052t)
To find the rate of change, we need to differentiate this equation with respect to time, t. The derivative of A with respect to t represents the rate of change or the derivative of A(T) with respect to T represents the rate of change of A with respect to T. However, in this case, we are interested in t years after 2000, so we will use the derivative with respect to t.
So, let's differentiate A = 1600e^(0.052t) with respect to t:
dA/dt = 0.052 * 1600 * e^(0.052t)
Now we have the derivative of A with respect to t, which represents the rate of change of the investment value. To find the annual increase as a percentage, we divide this rate of change by the initial investment value and multiply by 100:
Annual Increase (%) = (dA/dt) / A * 100
Plugging in the values:
Annual Increase (%) = (0.052 * 1600 * e^(0.052t)) / (1600e^(0.052t)) * 100
Simplifying the equation:
Annual Increase (%) = 0.052 * 100
Annual Increase (%) = 5.2%
Therefore, the account increases by approximately 5.2% each year.