Question
Olivia just graduated with an Associates degree and is working as a secretary in a firm earning $9.75
an hour. Her contract stipulates that she will get an annual raise of 6%
until she reaches the rate of $15.50
an hour. How long would it take for Olivia to earn $15.50
an hour?
an hour. Her contract stipulates that she will get an annual raise of 6%
until she reaches the rate of $15.50
an hour. How long would it take for Olivia to earn $15.50
an hour?
Answers
GPT-4o mini
To determine how long it will take for Olivia to reach an hourly wage of $15.50 from her current wage of $9.75 with a 6% annual raise, we can set up the following equation to model her wage over time:
1. **Current hourly wage**: \( P_0 = 9.75 \)
2. **Annual raise percentage**: \( r = 6\% = 0.06 \)
3. **Target wage**: \( P_t = 15.50 \)
The wage at the end of \( t \) years can be calculated using the formula:
\[
P_t = P_0 \times (1 + r)^t
\]
We want to find \( t \) such that:
\[
15.50 = 9.75 \times (1 + 0.06)^t
\]
Now, we can divide both sides by 9.75:
\[
\frac{15.50}{9.75} = (1.06)^t
\]
Calculating the left-hand side:
\[
\frac{15.50}{9.75} \approx 1.587179
\]
So we have:
\[
1.587179 = (1.06)^t
\]
To solve for \( t \), we will take the natural logarithm of both sides:
\[
\ln(1.587179) = \ln((1.06)^t)
\]
Using the properties of logarithms, this simplifies to:
\[
\ln(1.587179) = t \cdot \ln(1.06)
\]
Now we solve for \( t \):
\[
t = \frac{\ln(1.587179)}{\ln(1.06)}
\]
Calculating the natural logarithm values:
\[
\ln(1.587179) \approx 0.4585
\]
\[
\ln(1.06) \approx 0.0583
\]
Now, substituting back into the equation for \( t \):
\[
t \approx \frac{0.4585}{0.0583} \approx 7.85
\]
Since \( t \) represents the number of years, we round up because Olivia won’t reach her target wage until the complete year is finished. Thus, it will take approximately **8 years** for Olivia to reach an hourly wage of $15.50.
1. **Current hourly wage**: \( P_0 = 9.75 \)
2. **Annual raise percentage**: \( r = 6\% = 0.06 \)
3. **Target wage**: \( P_t = 15.50 \)
The wage at the end of \( t \) years can be calculated using the formula:
\[
P_t = P_0 \times (1 + r)^t
\]
We want to find \( t \) such that:
\[
15.50 = 9.75 \times (1 + 0.06)^t
\]
Now, we can divide both sides by 9.75:
\[
\frac{15.50}{9.75} = (1.06)^t
\]
Calculating the left-hand side:
\[
\frac{15.50}{9.75} \approx 1.587179
\]
So we have:
\[
1.587179 = (1.06)^t
\]
To solve for \( t \), we will take the natural logarithm of both sides:
\[
\ln(1.587179) = \ln((1.06)^t)
\]
Using the properties of logarithms, this simplifies to:
\[
\ln(1.587179) = t \cdot \ln(1.06)
\]
Now we solve for \( t \):
\[
t = \frac{\ln(1.587179)}{\ln(1.06)}
\]
Calculating the natural logarithm values:
\[
\ln(1.587179) \approx 0.4585
\]
\[
\ln(1.06) \approx 0.0583
\]
Now, substituting back into the equation for \( t \):
\[
t \approx \frac{0.4585}{0.0583} \approx 7.85
\]
Since \( t \) represents the number of years, we round up because Olivia won’t reach her target wage until the complete year is finished. Thus, it will take approximately **8 years** for Olivia to reach an hourly wage of $15.50.