Suppose that a new employee starts working at $7.76 per hour and receives a 4%raise each year. After time t, in years, his hourly wage is given by the equation y=$7.76(1.04)^t. Find the amount of time after which he will be earning $10.00 per hour.

Question

After what amount of time will the employee be earning $10.00 per hour?
____years(Round to the nearest tenth of a year as needed)

Answer 1

To determine how long it will take for the employee to be earning $10.00 per hour, we need to set the wage equation equal to $10.00 and solve for \( t \).

The equation is:

\[ y = 7.76(1.04)^t \]

We want to find \( t \) when \( y = 10.00 \). Setting up the equation:

\[ 10.00 = 7.76(1.04)^t \]

Now, divide both sides by 7.76:

\[ \frac{10.00}{7.76} = (1.04)^t \]

Calculating the left side:

\[ \frac{10.00}{7.76} \approx 1.2876 \]

So now we have:

\[ 1.2876 = (1.04)^t \]

Next, we take the logarithm of both sides to solve for \( t \):

\[ \log(1.2876) = t \cdot \log(1.04) \]

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

\[ t = \frac{\log(1.2876)}{\log(1.04)} \]

Calculating the logarithms:

Using a calculator:

• \( \log(1.2876) \approx 0.1107 \)
• \( \log(1.04) \approx 0.0170 \)

Now substitute these values back into the equation:

\[ t \approx \frac{0.1107}{0.0170} \approx 6.5 \]

So, the employee will be earning $10.00 per hour after approximately \( 6.5 \) years.

Rounding to the nearest tenth of a year:

Answer: 6.5 years