A: FUNCTION
Q1. (a) The function C (x) = 25x + 80,000 express the total cost C (x) (in K£) of
manufacturing x units of a product. If the maximum number of units which can be
produced equals 20,000, state the restricted domain and range for this function.
(b) An analysis of enrolment in MBA Parallel Programme indicates that enrolment will decline as the pool of University graduates in the country shrink. The estimates indicate that the number of applicants (y) for the coming year will behave according to the function
y = f (t) = 250 – 20t where t equals time in years measured from 1999 (t = 0)
(i) What is the expected number of applicants in 2002; 2010.
(ii) Do you think this function is accurate as a predictor indefinitely into the
future?
(c) Passengers’ volume on a small regional airline has been declining 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 years (t = 0 for 1981).
i) Determine the linear function n = (t).
ii) Interpret the meaning of the slope.
iii) What is the number of passengers expected to be in the year 2002?
iv) It is estimated that the airline go out of business if the volume falls below 180,000. According to your function in part (a), when will this happen.
Assume that a manufacturer can purchase a needed component from a supplier at a cost of $8 per unit, or it can invest $40,000 in equipment and produce the item at the cost of $5.50 per unit.
i) Determine the quantity for which total costs are equal for the make and buy alternatives.
ii) What is the minimum cost alternative if 15,000 units are required? What is the minimum cost?
Sollutions
Q1. (a) For the function C(x) = 25x + 80,000, the maximum number of units that can be produced is given as 20,000.
(i) The restricted domain would be the values of x that are less than or equal to 20,000. So, the restricted domain is x ≤ 20,000.
(ii) The range for this function will be the values of C(x). Since C(x) represents the total cost in K£, the range would be any value greater than or equal to 80,000 (the initial cost) multiplied by the number of units x. So, the range is C(x) ≥ 80,000x.
(b) For the function y = f(t) = 250 - 20t, where t represents the time in years measured from 1999 (t = 0):
(i) To find the expected number of applicants in 2002, substitute t = 2002 - 1999 = 3 into the function: y = 250 - 20(3) = 190. Therefore, the expected number of applicants in 2002 is 190.
To find the expected number of applicants in 2010, substitute t = 2010 - 1999 = 11 into the function: y = 250 - 20(11) = 50. Therefore, the expected number of applicants in 2010 is 50.
(ii) This function might not be accurate as a predictor indefinitely into the future. It assumes a linear decline in the number of applicants, but real-world factors like changes in the job market or educational policies could lead to different trends.
(c) For the linear decline in passengers on a small regional airline:
(i) To determine the linear function n = υ(t), we can use the two given data points: (1981, 245,000) and (1986, 215,000).
Using the slope formula, m = (215,000 - 245,000) / (1986 - 1981) = -6,000.
Using the point-slope form of a line, we have n - 245,000 = -6,000(t - 1981).
Simplifying the equation, we get n = -6,000t + 12,270,000.
Therefore, the linear function is n = υ(t) = -6,000t + 12,270,000.
(ii) The slope (-6,000) represents the rate at which the number of passengers is declining per year. In this case, it means that the number of passengers decreases by an average of 6,000 per year.
(iii) To find the number of passengers expected in the year 2002, substitute t = 2002 - 1981 = 21 into the function: n = -6,000(21) + 12,270,000 = 12,150,000. Therefore, the number of passengers expected in the year 2002 is 12,150,000.
(iv) To find when the volume falls below 180,000, we can set n = 180,000 in the function and solve for t:
180,000 = -6,000t + 12,270,000
-6,000t = -12,090,000
t = 2.015
Since t = 0 represents the year 1981, we can conclude that the airline is expected to go out of business in the year 1983.
For the make and buy alternatives:
(i) To find the quantity for which total costs are equal, we can set the two cost functions equal to each other:
8x = 40,000 + 5.50x
2.50x = 40,000
x = 16,000
Therefore, the quantity for which the total costs are equal for the make and buy alternatives is 16,000 units.
(ii) To find the minimum cost alternative if 15,000 units are required, we compare the costs of the make and buy alternatives.
For the "make" option, the total cost would be: 15,000 * 5.50 = $82,500.
For the "buy" option, the total cost would be: 15,000 * 8 = $120,000.
Therefore, the minimum cost alternative is the "make" option with a minimum cost of $82,500.