Question 1186380: Ridership on a small regional airline has been declining, approximately at a linear rate. In 1981 the number of passengers was 245,000; in 1986 the number was 215,000. If n equals the number of passengers using the airline per year and t equals time measured in year (t=0 for 1981)
Determine the linear estimating function n=f(t)
Interpret the meaning of the slope
What is the number of riders expected to equal in the year 2000?
It is estimated that the airline will go out of business if ridership falls below 180,000. According to your function in part a, when will this happen?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to solve this problem:
**1. Determine the linear estimating function n = f(t):**
We are given two points on the line:
* (t₁, n₁) = (0, 245000) (1981)
* (t₂, n₂) = (5, 215000) (1986)
First, calculate the slope (m):
m = (n₂ - n₁) / (t₂ - t₁)
m = (215000 - 245000) / (5 - 0)
m = -30000 / 5
m = -6000
Now, use the point-slope form of a linear equation with one of the points (e.g., (0, 245000)):
n - n₁ = m(t - t₁)
n - 245000 = -6000(t - 0)
n = -6000t + 245000
So, the linear estimating function is: **n(t) = -6000t + 245000**
**2. Interpret the meaning of the slope:**
The slope, -6000, represents the average *decrease* in the number of passengers per year. In other words, the airline's ridership is declining by approximately 6,000 passengers each year.
**3. What is the number of riders expected in the year 2000?**
The year 2000 is 19 years after 1981 (2000 - 1981 = 19), so t = 19. Plug this into the equation:
n(19) = -6000 * 19 + 245000
n(19) = -114000 + 245000
n(19) = 131000
Therefore, the number of riders expected in the year 2000 is **131,000**.
**4. When will ridership fall below 180,000?**
We need to find the value of t when n(t) is less than 180,000:
180000 = -6000t + 245000
-65000 = -6000t
t = -65000 / -6000
t ≈ 10.83
Since t represents the number of years after 1981, we add this to 1981:
1981 + 10.83 ≈ 1991.83
Since we're dealing with whole numbers of years, ridership will likely fall below 180,000 in the year **1991** or **1992**.
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