Question

Could you please help me with this? It’s part of my grandson’s homework!

Answer 1

Answer:

The salary after 2 years is $8.83 per hour.

The salary will be more than $10 per hour in 8 years.

Step-by-step explanation:

To calculate a 2.5% raise, we have to multiply the original salary by 1.025:

$8.40\times1.025$

For the second year, the salary would be:

$8.40\times1.025\times1.025$

For the third year, the salary would be:

$8.40\times1.025\times1.025\times1.025$

Notice that a pattern emerges: To calculate the salary on the n-th year, we have to multiply the original salary by 1.025 n times. In other words, the formula to calculate the salary for the n-th year is:

$8.40\times1.025^n$

This way, the salary after two years would be:

$8.40\times1.025^2=8.83$

$8.83

To find when will the salary be over $10 per hour, we equal our equation to 10, solve for n and ceil to the nearest integer, as following:

$\begin{gathered} 10=8.40\times1.025^n\rightarrow\frac{10}{8.40}=1.025^n\rightarrow\ln(\frac{10}{8.40})=\ln(1.025^n) \\ \\ \rightarrow\ln(\frac{10}{8.40})=n\ln(1.025) \\ \\ \rightarrow n=\frac{\ln(\frac{10}{8.40})}{\ln(1.025^)} \\ \\ \rightarrow n=7.06 \\ \Rightarrow n=8 \end{gathered}$

This way, the salary will be more than $10 per hour in 8 years.