An investment of $12,000 earns interest at an annual rate of 8% compounded continuously.
Question 1: Find the rate of change of the amount in the account after two years.
Question 2:Find the instantaneous rate of change of the amount in the account at the time the amount is equal to $12,000
A = 12000 e^0.08x
dA/dt = 960 e^.08x
Now you can answer the questions. Don't forget your Algebra II now that you're taking calculus ...
To answer both of these questions, we need to use the formula for continuously compounded interest, which is given by:
A = P * e^(rt)
Where:
A = the amount after time t
P = the principal (initial investment)
e = Euler’s number (approximately equal to 2.71828)
r = the annual interest rate
t = time in years
Question 1: Find the rate of change of the amount in the account after two years.
To find the rate of change of the amount after two years, we need to differentiate the formula with respect to time (t).
Differentiating both sides of the equation, we get:
dA/dt = P * r * e^(rt)
Plugging in the given values:
P = $12,000
r = 8% = 0.08
t = 2 years
We can now calculate the rate of change of the amount:
dA/dt = $12,000 * 0.08 * e^(0.08 * 2)
Using the value for e, we can simplify this expression and calculate the rate of change.
Question 2: Find the instantaneous rate of change of the amount in the account at the time the amount is equal to $12,000.
To find the instantaneous rate of change at a specific time, we need to calculate the derivative of the formula and plug in the values for the time when the amount is $12,000.
Differentiating both sides of the formula, we get:
dA/dt = P * r * e^(rt)
We know that A = $12,000, so we have:
$12,000 = P * e^(rt)
Now, we take the derivative of both sides with respect to time (t):
0 = P * r * e^(rt) * dt/dt + P * e^(rt) * dr/dt
Since we're interested in finding the instantaneous rate of change at the time the amount is equal to $12,000, we set this equation equal to zero and solve for dt/dt (instantaneous rate of change):
0 = P * r * e^(rt) * dt/dt + P * e^(rt) * dr/dt
Solve for dt/dt to find the instantaneous rate of change at the time the amount is equal to $12,000.