To find the total income produced by a continuous income stream over a specific time period, we analyze the given function [tex]\( f(t) \)[/tex]. In this situation, the flow rate [tex]\( f(t) \)[/tex] is constant at 4000 units per year.
Step-by-Step Solution:
1. Understand the Function:
The rate of flow [tex]\( f(t) = 4000 \)[/tex] indicates that, at any given year [tex]\( t \)[/tex], the income flow rate remains constant at 4000 units per year.
2. Determine the Time Period:
We need to calculate the total income over the first 5 years. So, our time period [tex]\( T \)[/tex] is from [tex]\( t = 0 \)[/tex] to [tex]\( t = 5 \)[/tex] years.
3. Calculate the Total Income:
Since the rate of flow is constant, the total income over the given period can be found by multiplying the rate of flow by the length of the time period.
- The rate of flow [tex]\( r \)[/tex] is 4000 units per year.
- The time period [tex]\( T \)[/tex] is 5 years.
Therefore:
[tex]\[
\text{Total Income} = \text{Rate of Flow} \times \text{Time Period}
\][/tex]
Plugging in the values:
[tex]\[
\text{Total Income} = 4000 \, \text{units/year} \times 5 \, \text{years}
\][/tex]
4. Perform the Multiplication:
[tex]\[
\text{Total Income} = 4000 \times 5 = 20000 \, \text{units}
\][/tex]
Therefore, the total income produced by the continuous income stream over the first 5 years is 20,000 units.
