A deposit of $5,000 at 4% interest compounded continuously will grow to V(t) = 5000e^0.04t dollars after t years. Find the average value during the first 20 years (that is, from time 0 to time 20). (Round your answer to the nearest cent.)
To find the average value during the first 20 years, we need to calculate the integral of V(t) from 0 to 20 and divide it by the length of the interval (20 years).
The integral of V(t) = 5000e^0.04t from 0 to 20 is given by:
∫[0,20] 5000e^0.04t dt
To integrate this expression, we can use the power rule of integration for exponential functions.
The integral of e^bt dt is equal to (1/b)e^bt + C, where C is the constant of integration.
Applying the power rule, we can find the integral of V(t):
∫[0,20] 5000e^0.04t dt = (5000/0.04)e^0.04t | [0,20]
Substituting the limits of integration:
= 125000e^0.04(20) - 125000e^0.04(0)
Simplifying:
= 125000(e^(0.8) - 1)
Calculating:
= 125000(2.22554 - 1)
= 125000(1.22554)
= 153192.5
To find the average value during the first 20 years, we divide the integral value by the length of the interval (20 years):
Average value = 153192.5 / 20
= 7659.625
Rounded to the nearest cent, the average value during the first 20 years is approximately $7659.63.
To find the average value during the first 20 years, we need to calculate the definite integral of the function V(t) = 5000e^0.04t over the interval [0, 20] and then divide it by the length of the interval.
The definite integral of V(t) = 5000e^0.04t is given by:
∫[0, 20] 5000e^0.04t dt
To evaluate this integral, we can use the properties of exponential functions and the power rule of integration.
Applying the power rule, the integral becomes:
= (5000/0.04) ∫[0, 20] e^0.04t dt
= (125,000) ∫[0, 20] e^0.04t dt
Next, using the property that ∫e^kt dt = (1/k)e^kt + C, where C is the constant of integration, we get:
= (125,000) [(1/0.04)e^0.04t] [from 0 to 20]
= (125,000) [(1/0.04)(e^0.04(20) - e^0.04(0))]
Simplifying further, we have:
= 125,000 [25e^0.8 - 1]
Calculating this expression will give us the total value of the investment over the first 20 years.
After obtaining the value, we can then divide it by the length of the interval, which is 20 years, to find the average value.
Lastly, to round the answer to the nearest cent, we can round the final result to two decimal places.
amount at the beginning = 5000
amount after 20 years
= 5000 e^(.04(20)) = 11127.70
evaluated the "average value" as defined in your text book or notes.
I don't know how it was defined for your course.