The following table gives the percentage, P, of households with a television set that also have a VCR. (Unlike the data in your textbook, this data is ficticious).
Year 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991
% having VCR 0.2 0.3 0.6 1.2 2.5 4.6 8.1 15.6 24.0 35.6 41.7 49.8 58.9 60.6
(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 = 62/1+370 e^−0.68 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?
The exact difference is 0.4 percent.
(a) During what year does the point of "diminishing returns" (i.e., the inflection point) appear to take place?
During the year when people started realizing that watching reruns of "The Golden Girls" on VCR wasn't as exciting as they initially thought.
(b) If the best fitting logistic function for this data is P = 62/1+370e^−0.68t, what is the limiting value (as t gets very large)?
As t gets very large, the limiting value would be the percentage of households with a television set that also have a VCR when people finally realize that streaming services are just easier and better.
(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?
The exact difference between the predicted value and the value stated in the table for the year 1991 is that the predicted value doesn't take into account the number of people who forgot to rewind their VHS tapes before returning them to the rental store. Trust me, that's a big difference.
(a) To determine the point of "diminishing returns" or the inflection point, we need to identify the year where the rate of increase in the percentage of households with a VCR starts to slow down significantly.
Looking at the data, we can observe that the rate of increase is somewhat steady until around 1986. After that, the percentage starts to increase at a much faster rate. So, the point of diminishing returns appears to take place around 1986.
(b) The limiting value is the maximum percentage that the logistic function approaches as t (years since 1978) gets very large.
The given logistic function is:
P = 62 / (1 + 370 * e^(-0.68t))
As t gets very large, the exponential term e^(-0.68t) approaches 0. Thus, the limiting value can be found by evaluating the function when t approaches infinity.
lim P(t) as t approaches infinity = 62 / (1 + 0) = 62 percent
Therefore, the limiting value, as t gets very large, is 62 percent.
(c) To find the exact difference between the prediction from the given function and the value stated in the table for the year 1991, we'll substitute t = 1991 - 1978 = 13 into the logistic function and compare it with the corresponding value from the table.
Using t = 13 in the logistic function:
P = 62 / (1 + 370 * e^(-0.68 * 13))
P ≈ 62.00 percent (rounded to two decimal places)
From the table, the percentage stated for the year 1991 is 58.9 percent.
Therefore, the exact difference between the predicted value from the given function and the value in the table for the year 1991 is:
58.9 - 62.00 ≈ -3.10 percent (rounded to two decimal places). This indicates that the predicted value from the function is 3.10 percent higher than the value from the table.
(a) To find the point of "diminishing returns" or the inflection point, we need to identify the year where the rate of increase in percentage having a VCR starts to slow down. We can observe the data in the table and look for a year where the percentage increase is not as significant as in previous years.
To visualize this, plot the percentage of households with a VCR against the years. You can use a graphing software or draw a graph manually. On the x-axis, represent the years (starting from 1978) and on the y-axis, represent the percentage values.
Plot the given data points from the table on the graph. Connect the points to see the trend of the data over time. Look for a point where the curve starts to flatten out. This point represents the inflection point or the year of diminishing returns.
(b) The logistic function given is P = 62/(1+370e^(-0.68t)), where t is the number of years since 1978. This is a function that models the percentage of households with a VCR over time.
To determine the limiting value as t gets very large, we need to find the value of P when t approaches infinity. As t gets very large, the term e^(-0.68t) approaches 0, since the exponent becomes increasingly negative.
Thus, the limiting value of P is equal to 62/(1+0) = 62 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 t = 1991 - 1978 = 13 into the logistic function and compare the result with the value stated in the table for the year 1991 (P = 60.6 percent).
Plug in the value of t into the logistic function: P = 62/(1+370e^(-0.68*13))
Evaluate the function using a calculator or software to find the predicted value for the year 1991.
Then, subtract the predicted value from the value stated in the table for 1991 to find the exact difference.