Year
1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991
% having VCR
0.2 0.3 0.7 1.2 2.4 4.6 8.7 15.5 23.9 34.2 45.4 50.6 56.6 61.3
(a) During what year does the point of "diminishing returns" (i.e., the inflection point) appear to take place?
During the year .
(b) If the best fitting logistic function for this data is
P = 64
1+380 eā0.68 t
,
(where t is years since 1978) what is the limiting value (as t gets very large)?
percent
(c) What is the exact difference (if any) between the value predicted by the given function and the value stated in the table for the year 1991?
(a) To determine the year when the point of "diminishing returns" or the inflection point appears, we need to look for the year where the rate of increase in percentage having VCR starts to slow down significantly.
From the given table, we can observe that the rate of increase is highest during the early years (1978-1980) and gradually slows down over time. In the later years (1988-1991), we can see that the rate of increase is relatively lower compared to earlier years.
To find the inflection point, we look for the year where the rate of increase starts to decrease significantly. In this case, it looks like the inflection point occurs around the year 1990. From this year onwards, the rate of increase in the percentage having VCR starts to slow down significantly.
Therefore, the year when the point of "diminishing returns" appears to take place is 1990.
(b) The best fitting logistic function for the given data is provided as:
P = 64 / (1 + 380 * e^(-0.68 * t))
In this function, "P" represents the percentage having VCR, "t" represents the number of years since 1978, and "e" is the mathematical constant approximately equal to 2.71828.
To find the limiting value of the logistic function as "t" gets very large, we need to evaluate the expression when "t" approaches infinity.
As "t" gets very large, the value of the exponential term -0.68 * t becomes infinitely small, approaching zero. Therefore, we can say that e^(-0.68 * t) approaches 0 as "t" tends to infinity.
When the exponential term approaches 0, the denominator of the logistic function becomes very close to 1. So, the limiting value of the logistic function as "t" gets very large is:
P = 64 / (1 + 380 * 0)
P = 64 / 1
P = 64 percent
Therefore, the limiting value of the logistic function as "t" gets very large is 64 percent.
(c) To find the exact difference between the value predicted by the given function and the value stated in the table for the year 1991, we need to substitute the value of "t" as 1991 - 1978 = 13 into the logistic function and calculate the difference.
Using the logistic function: P = 64 / (1 + 380 * e^(-0.68 * t))
Substituting t = 13 into the function:
P = 64 / (1 + 380 * e^(-0.68 * 13))
Calculating the value of the function for t = 13:
P ā 64 / (1 + 380 * e^(-8.84))
Now, we compare this calculated value with the value mentioned in the table for the year 1991, which is 61.3 percent.
The difference between the value predicted by the function and the value stated in the table for the year 1991 can be found by taking the absolute difference:
Difference = |Calculated value - Value from table|
Difference = |64 / (1 + 380 * e^(-8.84)) - 61.3|
Calculating this difference will give you the exact difference between the predicted value and the value from the table for the year 1991.