Question

Suppose that a new employee starts working at ​$7.85 per hour and receives a 5​% raise each year. After time​ t, in​ years, his hourly wage is given by the equation y equals $ 7.85 left parenthesis 1.05 right parenthesis Superscript t Baseline . Find the amount of time after which he will be earning​ $10.00 per hour.

Answer 1

To find the time \( t \) after which the employee will be earning $10.00 per hour, we can use the given equation for the hourly wage:

\[
y = 7.85 (1.05)^t
\]

We want to find \( t \) when \( y = 10.00 \):

\[
10.00 = 7.85 (1.05)^t
\]

Now, we'll divide both sides by 7.85 to isolate \( (1.05)^t \):

\[
\frac{10.00}{7.85} = (1.05)^t
\]

Now calculate \( \frac{10.00}{7.85} \):

\[
\frac{10.00}{7.85} \approx 1.2738
\]

So we have:

\[
1.2738 = (1.05)^t
\]

To solve for \( t \), we can take the logarithm of both sides:

\[
\log(1.2738) = \log((1.05)^t)
\]

Using the property of logarithms that allows us to bring the exponent down, we get:

\[
\log(1.2738) = t \cdot \log(1.05)
\]

Now, we can solve for \( t \):

\[
t = \frac{\log(1.2738)}{\log(1.05)}
\]

Calculating the logarithms:

- \( \log(1.2738) \approx 0.1053 \)
- \( \log(1.05) \approx 0.0212 \)

Now we can calculate \( t \):

\[
t \approx \frac{0.1053}{0.0212} \approx 4.97
\]

Thus, it will take approximately **5 years** for the employee to earn $10.00 per hour.