The amount of the salary that Herbert agreed was $40836 per annum approx approximately. In his first 10 years, he earned approx $524,384
If the initial amount (also called as principal amount) is P, and the interest rate is R% per unit time, and it is left for T unit of time for that compound interest, then the interest amount earned is given by:
[tex]CI = P(1 +\dfrac{R}{100})^T - P[/tex]
The final amount becomes:
[tex]A = CI + P\\A = P(1 +\dfrac{R}{100})^T[/tex]
Let the initial salary that Herbert agreed on was 'x' dollars annually.
We can use compound interest formula as there is compounding of some amount at some rate for some time, where interest earned will correspond to the increment in salary.
Then, as the rate of increment is 4.5% per year, and time is of 10 years, so compounding the incremented amount on the base principal amount gives final salary at the 10th year as:
[tex]A = x(1 +\dfrac{4.5}{100})^{10}\\\\63417 = x(1+0.045)^{10}\\\\x \approx \dfrac{63417}{(1.045^{10})} \approx 40836 \: \text{\: (in dollars)}[/tex]
$40836 was his first year's salary.
For Tth year's salary, we just need to replace 10 with T.
Thus, sum of the money he earned in 10 years will be:
[tex]S = \sum A_i = \sum_{i=1}^{10} 40836(1.045)^i\\S \approx 42674 +44594+46601 +48698 +50889+53179 +55572 +58073 +60687 + 63417\\S \approx 524384[/tex]
Thus, the amount of the salary that Herbert agreed was $40836 per annum approx approximately. In his first 10 years, he earned approx $524,384
Learn more about compound interest here:
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