To estimate Korey's expected profits for the next year, we first need to determine the growth rates for each year and then find the average growth rate. Following this, we will use the average growth rate to project the profits for the next year.
Step 1: Calculate the Growth Rates
1. Growth rate from Year 1 to Year 2:
The growth rate is calculated as:
[tex]\[
\text{Growth Rate}_1 = \frac{\text{Profits in Year 2} - \text{Profits in Year 1}}{\text{Profits in Year 1}}
\][/tex]
Given:
[tex]\[
\text{Profits in Year 1} = \$14,250.00
\][/tex]
[tex]\[
\text{Profits in Year 2} = \$15,390.00
\][/tex]
Substituting the values:
[tex]\[
\text{Growth Rate}_1 = \frac{\[tex]$15,390.00 - \$[/tex]14,250.00}{\[tex]$14,250.00} = \frac{\$[/tex]1,140.00}{\$14,250.00} \approx 0.08
\][/tex]
2. Growth rate from Year 2 to Year 3:
Similarly, we calculate:
[tex]\[
\text{Growth Rate}_2 = \frac{\text{Profits in Year 3} - \text{Profits in Year 2}}{\text{Profits in Year 2}}
\][/tex]
Given:
[tex]\[
\text{Profits in Year 3} = \$16,621.20
\][/tex]
Substituting the values:
[tex]\[
\text{Growth Rate}_2 = \frac{\[tex]$16,621.20 - \$[/tex]15,390.00}{\[tex]$15,390.00} = \frac{\$[/tex]1,231.20}{\$15,390.00} \approx 0.08
\][/tex]
3. Growth rate from Year 3 to Year 4:
Similarly, we calculate:
[tex]\[
\text{Growth Rate}_3 = \frac{\text{Profits in Year 4} - \text{Profits in Year 3}}{\text{Profits in Year 3}}
\][/tex]
Given:
[tex]\[
\text{Profits in Year 4} = \$17,950.90
\][/tex]
Substituting the values:
[tex]\[
\text{Growth Rate}_3 = \frac{\[tex]$17,950.90 - \$[/tex]16,621.20}{\[tex]$16,621.20} = \frac{\$[/tex]1,329.70}{\$16,621.20} \approx 0.08
\][/tex]
Step 2: Calculate the Average Growth Rate
The average growth rate can be found by taking the sum of the individual growth rates and dividing by the number of years:
[tex]\[
\text{Average Growth Rate} = \frac{\text{Growth Rate}_1 + \text{Growth Rate}_2 + \text{Growth Rate}_3}{3} \approx \frac{0.08 + 0.08 + 0.08}{3} \approx 0.08
\][/tex]
Step 3: Estimate the Profits for Year 5
Using the average growth rate, we can estimate the profits for Year 5:
[tex]\[
\text{Estimated Profits in Year 5} = \text{Profits in Year 4} \times (1 + \text{Average Growth Rate})
\][/tex]
Given:
[tex]\[
\text{Profits in Year 4} = \$17,950.90
\][/tex]
Substituting the values:
[tex]\[
\text{Estimated Profits in Year 5} = \[tex]$17,950.90 \times (1 + 0.08) = \$[/tex]17,950.90 \times 1.08 \approx \$19,386.97
\][/tex]
Therefore, the best estimate for Korey to quote as expected profits in his new business plan for the next year is:
[tex]\[
\boxed{\$19,386.97}
\][/tex]
