Willow is in her fourth year working for the same company. Her starting salary was $48,000. The next year, her salary was $50,400. The following year, her salary was $52,920. This year, her salary is $55,566. Willow’s boss says her salary will continue to increase by the same percentage each year.

Question

Write the values given in the scenario as a sequence.  48,000, 50,400, 52,920, 55,566,
create a function that models this sequence/ scenario
By what percentage is Willow's salary increasing each year ?show your work?
Well I wants to use your function to predict your salary in the future she wants to make sure her predictions are made to reasonable level of accuracy how can she do this?

Answer 1

To analyze Willow's salary increases, we can express her salary growth as a sequence of values:

1. Year 1: $48,000
2. Year 2: $50,400
3. Year 3: $52,920
4. Year 4: $55,566

Identify the Pattern and Model the Sequence

We notice that the salaries are increasing each year by a certain percentage. To express this as a function, we first need to determine the percentage increase.

Let's calculate the percentage increase from each year's salary to the next:

Step 1: Calculate the percentage increase

To find the percentage increase from year to year, we can use the formula:

\[ \text{Percentage Increase} = \left(\frac{\text{new salary} - \text{old salary}}{\text{old salary}}\right) \times 100 \]

• From Year 1 to Year 2: \[ \frac{50,400 - 48,000}{48,000} \times 100 = \frac{2,400}{48,000} \times 100 = 5% \]

• From Year 2 to Year 3: \[ \frac{52,920 - 50,400}{50,400} \times 100 = \frac{2,520}{50,400} \times 100 \approx 5% \]

• From Year 3 to Year 4: \[ \frac{55,566 - 52,920}{52,920} \times 100 = \frac{2,646}{52,920} \times 100 \approx 5% \]

The percentage increase is consistent at approximately 5% per year.

Step 2: Model the Sequence with a Function

We can express Willow's salary as a function of the year \( n \), where \( n = 0 \) corresponds to the first year of her employment:

\[ S(n) = S_0 \cdot (1 + r)^n \]

Where:

• \( S(n) \) is the salary in year \( n \).
• \( S_0 \) is the starting salary ($48,000).
• \( r \) is the percentage increase (5% or 0.05).
• \( n \) is the number of years since starting (0, 1, 2, ...).

Substituting the values we have:

\[ S(n) = 48000 \cdot (1 + 0.05)^n \]

Step 3: Predict Future Salaries

To predict Willow's salary in the future, we can plug in the desired value of \( n \) into the formula \( S(n) \):

For example, to predict her salary in year 5:

• \( n = 4 \): \[ S(4) = 48000 \cdot (1.05)^4 \]

Calculating \( S(4) \): \[ S(4) = 48000 \cdot (1.21550625) \approx 58244.19 \]

Willow's predicted salary in year 5 (the fifth year of employment) will be approximately $58,244.19.

Ensuring Reasonable Accuracy in Predictions

To ensure that predictions are made with a reasonable level of accuracy, Willow should:

1. Verify Consistency: Check if the percentage increase remains constant over time. If it appears to fluctuate, she may need to adjust her predictions accordingly.

2. Periodical Reviews: Update the model occasionally based on actual salary changes provided by her employer.

3. Consider Company Policy: Take into account any company-wide pay adjustments, inflation factors, or performance evaluations that could affect her salary.

By incorporating these considerations into her model, Willow can make more accurate and reliable predictions about her future salary.