Alright, let's break it down step by step.
The math teacher earned $48,000 in her first year. She then gets a 5% raise each year after that. This is an example of what we call a "geometric sequence" in mathematics, where each term is a fixed percentage of the term before it.
Let's first figure out how much she earned in her 10th year.
To calculate her salary for a given year, we can use the formula:
Salary = Initial Salary * (1 + raise rate)^(number of years - 1)
For the 10th year, the number of years is 10, the initial salary is $48,000, and the raise rate is 5%, which is equal to 0.05 when expressed as a decimal.
Plugging these values into the formula:
Salary in 10th year = $48,000 * (1 + 0.05)^(10 - 1)
= $48,000 * (1.05)^9
= $48,000 * 1.5513 (rounded to four decimal places)
≈ $74,461.44
So, in her 10th year, she earned approximately $74,461.44.
Now, let's calculate the total earnings over the 10-year period.
To do this, we can use the formula for the sum of a geometric sequence:
Total Earnings = Initial Salary * ( (1 + raise rate)^number of years - 1 ) / raise rate
Plugging in the values:
Total Earnings = $48,000 * ( (1.05)^10 - 1 ) / 0.05
= $48,000 * (1.6289 - 1) / 0.05 (rounded to four decimal places)
= $48,000 * 12.5778 (rounded to four decimal places)
≈ $603,734.40
So, over a 10-year period, she earned approximately $603,734.40.
I hope that helps! Let me know if you have any more questions or need further clarification on anything.