Sales of Version 5.0 of a computer software package start out high and decrease exponentially. At time t, in years, the sales are s(t)=20e^-t thousands of dollars per year. After 2 years, Version 6.0 of the software is released and replaces Version 5.0. Assume that all income from software sales is immediately invested in government bonds which pay interest at a 7 percent rate compounded continuously, calculate the total value of sales of Version 5.0 over the two year period.
Any ideas??
To calculate the total value of sales of Version 5.0 over the two-year period, you need to integrate the sales function over the interval of 0 to 2 years.
The sales function is given as s(t) = 20e^(-t) thousands of dollars per year.
To perform the integration, follow these steps:
1. Integrate the sales function over the interval [0, 2]:
∫[0,2] 20e^(-t) dt
2. To integrate, use the integration rule for exponential functions:
∫e^(a*t) dt = (1/a) * e^(a*t) + C
Applying this rule, the integral becomes:
∫20e^(-t) dt = (-20) * e^(-t) + C
3. Evaluate the definite integral over the interval [0, 2]:
[(-20) * e^(-t)] evaluated from 0 to 2
Plugging in the upper and lower limits of integration, the expression becomes:
[(-20) * e^(-2)] - [(-20) * e^(-0)]
4. Simplify the expression:
[(-20) * e^(-2)] - [(-20) * e^0] = (-20) * (e^0 - e^(-2))
5. Calculate the numerical value of the expression using a calculator:
(-20) * (e^0 - e^(-2)) ≈ 7.877
Therefore, the total value of sales of Version 5.0 over the two-year period is approximately 7.877 thousand dollars.