A fixed amount 15000 is invested in a tax-sheltered account at an annual rate of percent, compounded continuously. Let represent the amount in the account at time .
Part A. Let percent. Give the differential equation that describes the rate of change in the amount with respect to time
To find the differential equation that describes the rate of change in the amount with respect to time, we need to use the formula for continuously compounded interest.
The formula for continuously compounded interest is given by:
A = P * e^(rt)
Where:
A = the amount in the account at time t
P = the principal amount (initial investment)
r = annual interest rate (as a decimal)
t = time in years
e = Euler's number (approximately 2.71828)
In this case, the interest rate is given as a percentage. To convert it to a decimal, we divide by 100. So if the interest rate is "k" percent, then r = k/100.
Let's use this information to find the differential equation.
Given:
Principal amount, P = $15000
Annual interest rate, r = k/100 (as a decimal)
We want to find the rate of change in the amount with respect to time, dA/dt.
To do this, we take the derivative of both sides of the equation A = P * e^(rt) with respect to time t:
dA/dt = P * r * e^(rt)
Substitute the values we have:
dA/dt = 15000 * (k/100) * e^((k/100)t)
Therefore, the differential equation that describes the rate of change in the amount with respect to time is:
dA/dt = 15000 * (k/100) * e^((k/100)t)