Obtain exponential functions in the form f(t) = Aert, if f(t) is the value after t years of a $9,000 investment depreciating continuously at an annual rate of 8.5%.
f(t) =
1
f(t)=Ae^rt
t=1, r= .085, A=9000
f(t) = 9000*e^(-0.085t)
To obtain the exponential function in the form f(t) = Aert, we need to find the value of A and r from the given information.
The formula for continuous depreciation is given by f(t) = P * e^(-rt), where P is the initial investment, r is the annual depreciation rate, and t is the number of years.
In this case, the initial investment is $9,000 and the annual depreciation rate is 8.5%, which can be written as 0.085 in decimal form.
So, we have the equation f(t) = 9000 * e^(-0.085t).
Thus, the exponential function is f(t) = 9000 * e^(-0.085t).
To obtain the exponential function in the form f(t) = Aert, we need to determine the values of A and r.
In this case, we are given that the investment depreciates continuously at an annual rate of 8.5%. Depreciating continuously means that the depreciation is happening constantly over time.
The formula for continuous depreciation or growth is given by:
f(t) = P * e^(rt)
Where:
f(t) is the value after t years
P is the initial investment (or principal amount)
e is Euler's number (approximately 2.71828)
r is the annual rate (as a decimal)
t is the time in years
We are given that the initial investment is $9,000. Therefore, P = $9,000.
The annual rate is given as 8.5%, which we need to convert to a decimal by dividing it by 100:
r = 8.5% / 100 = 0.085
Substituting these values into the formula, we get:
f(t) = $9,000 * e^(0.085t)
Thus, the exponential function representing the value of the investment after t years is:
f(t) = $9,000 * e^(0.085t)