题目信息
| x | C(x) |
| 0 | 25,000 |
| 10 | 24,919 |
| 20 | 24,846 |
| 30 | 24,781 |
| 40 | 24,724 |
| 50 | 24,675 |
A certain manufacturer uses the function C(x) = 0.04x2 – 8.5x + 25,000 to calculate the cost, in dollars, of producing x thousand units of its product. The table above gives values of this cost function for values of x between 0 and 50 in increments of 10. For which of the following intervals is the average rate of decrease in cost less than the average rate of decrease in cost for each of the other intervals?
A:From x = 0 to x = 10
B:From x = 10 to x = 20
C:From x = 20 to x = 30
D:From x = 30 to x = 40
E:From x = 40 to x = 50
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正确答案：
E:From x = 40 to x = 50
Arithmetic Applied problems
Since the average rate of decrease of C(x) in the interval from x = a to x = a + 10 is
, we are to determine for which value of a, chosen from the numbers 0, 10, 20, 30, and 40, the magnitude of 
is the least, or equivalently, for which of these values of a the magnitude of C(a + 10) − C(a) is the least. Probably the most straightforward method is to simply calculate or estimate the difference C(a + 10) – C(a) for each of these values of a, as shown in the table below.
Alternatively, since the graph of C(x) = 0.04x2 – 8.5x + 25,000 is a parabola with vertex at x =
= 100, it follows that the graph levels out as the value of x approaches a number that is approximately equal to 100. Therefore, among the intervals given, the least magnitude in the average rate of change of C(x) occurs for the interval closest to the vertex, which is the interval from x = 40 to x = 50.
The correct answer is E.
Since the average rate of decrease of C(x) in the interval from x = a to x = a + 10 is
, we are to determine for which value of a, chosen from the numbers 0, 10, 20, 30, and 40, the magnitude of is the least, or equivalently, for which of these values of a the magnitude of C(a + 10) − C(a) is the least. Probably the most straightforward method is to simply calculate or estimate the difference C(a + 10) – C(a) for each of these values of a, as shown in the table below.| a to a + 10 | C(a + 10) – C(a) |
| 0 to 10 | –81 |
| 10 to 20 | –73 |
| 20 to 30 | –65 |
| 30 to 40 | –57 |
| 40 to 50 | –49 |
Alternatively, since the graph of C(x) = 0.04x2 – 8.5x + 25,000 is a parabola with vertex at x =
= 100, it follows that the graph levels out as the value of x approaches a number that is approximately equal to 100. Therefore, among the intervals given, the least magnitude in the average rate of change of C(x) occurs for the interval closest to the vertex, which is the interval from x = 40 to x = 50.The correct answer is E.
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