To determine the month when the revenue will equal the expenses, we will follow and explain each step carefully:
1. Identify known upfront data:
- Revenue data over months:
- January: [tex]$4000
- February: $[/tex]9000
- March: [tex]$13000
- April: $[/tex]16000
- May: [tex]$20500
- Expenses data over months:
- January: $[/tex]22000
- February: [tex]$24000
- March: $[/tex]25000
- April: [tex]$27000
- May: $[/tex]30000
2. Determine the linear models for both revenue and expenses:
- Model for revenue: [tex]\( \text{Revenue} = 4000.0 \times (\text{Month}) + 500.0 \)[/tex]
- Model for expenses: [tex]\( \text{Expenses} = 1900.0 \times (\text{Month}) + 19900.0 \)[/tex]
3. Equilibrium month calculation:
Set the two equations equal to each other to find the point where revenue equals expenses:
[tex]\[4000.0 \times \text{Month} + 500.0 = 1900.0 \times \text{Month} + 19900.0\][/tex]
4. Simplify and solve for the month:
[tex]\[
4000.0 \times \text{Month} - 1900.0 \times \text{Month} = 19900.0 - 500.0 \\
2100.0 \times \text{Month} = 19400.0 \\
\text{Month} = \frac{19400.0}{2100.0} = 9.238095238095237
\][/tex]
5. Interpret the fractional month value:
The equilibrium point is at month 9.238, which means the revenue is predicted to equal the expenses sometime after the 9th month.
6. Determine the specific month:
The given options are October (10th month) and September (9th month). Since the equilibrium month value (9.238) is slightly after the 9th month, it falls in early October.
Therefore, the correct prediction for the month in which revenue equals expenses is October.
