Final answer:
To find when the average salary of public school teachers reached $44,000, we set up the equation 44 = -1.562 + 14.584ln(t) and solved for t, leading to the conclusion that the salary reached $44,000 approximately in the year 2003.
Step-by-step explanation:
The student's question is asking to determine the year when the average salary of public school teachers reached $44,000 based on the given model. We are given a logarithmic equation, y=-1.562+14.584ln(t), where y represents the salary in thousands of dollars and t represents the time with t=5 corresponding to 1985.
To find when the salary reached $44,000, we first need to convert $44,000 to thousands of dollars, which would be 44 (since we're working in thousands). Then we'll set up the equation 44 = -1.562 + 14.584ln(t) and solve for t.
- Convert the target salary to the unit used in the equation: $44,000 → 44 thousand dollars.
- Plug into the equation: 44 = -1.562 + 14.584ln(t).
- Solve for t: First, isolate the logarithm by moving -1.562 to the other side. Then divide by 14.584 to isolate ln(t). To find t, take the exponent of the resulting number using the base e.
Let's perform the calculation:
- 44 + 1.562 = 14.584ln(t)
- 45.562 / 14.584 = ln(t)
- ln(t) = 3.124
- t = e3.124
- t ≈ 22.75
Since t=5 represents 1985, and each unit increase in t represents one year, t=22.75 would roughly correspond to the year 1985 + 22.75 - 5 = 2002.75, which we could round to the year 2003.
Therefore, in the year 2003, the average salary for public school teachers was approximately $44,000 according to the model provided.