The number of customers that visit a local small business is 37,800 and has been continuously declining at a rate of 1.7% each year. What is the approximate number of customers that visit the business in 12 years?
The population P after t years is evidently
P(t) = 37800 * 0.983^t
so, plug in t=12
To find the approximate number of customers that will visit the business in 12 years, we need to calculate the declining number of customers each year.
Step 1: Calculate the annual decline rate:
- Decline rate = 1.7% = 0.017 (as a decimal)
Step 2: Calculate the number of customers each year:
- The number of customers in the first year = 37,800
- The number of customers in the second year = 37,800 - (0.017 * 37,800)
- The number of customers in the third year = (37,800 - (0.017 * 37,800)) - (0.017 * (37,800 - (0.017 * 37,800)))
- Continue this pattern for 12 years, substituting the previous year's number of customers into the formula for each subsequent year.
Step 3: Calculate the number of customers in the 12th year:
- Substitute the previous year's number of customers into the formula for the 12th year.
Using a programming or spreadsheet software, we can calculate the approximate number of customers visiting the business in 12 years.
To find the approximate number of customers that will visit the business in 12 years, we can calculate the future value using compound interest formula:
Future Value = Present Value * (1 + interest rate)^number of years
In this case:
Present Value (P) = 37,800 (number of customers currently)
Interest Rate (r) = -1.7% = -0.017 (since the number of customers is declining at a rate of 1.7% each year)
Number of years (t) = 12 (as we are considering the customers in 12 years)
Now let's substitute these values into the formula and calculate the future value:
Future Value = 37,800 * (1 - 0.017)^12
Future Value = 37,800 * (0.983)^12
Future Value ≈ 37,800 * 0.756
Future Value ≈ 28,573.68
Therefore, the approximate number of customers that will visit the business in 12 years is approximately 28,574.