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The question in the image involves determining the time $ t $ in years for an employee’s hourly wage to reach $10.00, starting from an initial wage of $7.62 with an annual raise of 4%.

The wage after $ t $ years is given by:
\[
y = 7.62 \times (1.04)^t
\]
where $ y $ is the hourly wage after $ t $ years.

We need to solve for $ t $ when $ y = 10.00 $.

1. Substitute $ y = 10.00 $ into the equation:
   \[
   10 = 7.62 \times (1.04)^t
   \]

2. Divide both sides by 7.62 to isolate the exponential term:
   \[
   \frac{10}{7.62} = (1.04)^t
   \]
   Calculating this, we get:
   \[
   1.312 \approx (1.04)^t
   \]

3. Take the natural logarithm of both sides to solve for $ t $:
   \[
   \ln(1.312) = t \cdot \ln(1.04)
   \]

4. Now, divide both sides by $ \ln(1.04) $:
   \[
   t = \frac{\ln(1.312)}{\ln(1.04)}
   \]

5. Calculate the values:
   - $ \ln(1.312) \approx 0.272 $
   - $ \ln(1.04) \approx 0.0392 $

   Thus,
   \[
   t \approx \frac{0.272}{0.0392} \approx 6.94
   \]

So, the employee will be earning $10.00 per hour after approximately **6.9 years** (rounded to the nearest tenth of a year).

Would you like more details on any step, or have any other questions?

### Related Questions
1. What would the wage be after 10 years if the raise rate was increased to 5%?
2. How many years would it take to reach $15.00 per hour with the same conditions?
3. If the initial wage were $8.00, how would that affect the time required to reach $10.00?
4. What is the effect on the result if the raise rate were compounded monthly rather than annually?
5. How would a one-time bonus at the end of the first year affect the time calculation?

#### Tip:
When solving for exponential growth in finance, logarithms are essential for isolating the exponent and finding the time needed to reach a target amount.

Suppose that a new employee starts working at $7.62 per hour and receives a 4% raise each year. After time t, in years, his hourly wage is given by the equation y = 7.62(1.04)^t. Find the amount of time after which he will be earning $10.00 per hour. Round to the nearest tenth of a year as needed.

Solve an exponential growth problem to find the time needed for an employee’s hourly wage to reach $10, given an initial wage of $7.62 with a 4% annual raise. This solution involves logarithmic calculations suitable for Grades 10-12.

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