20. A company produces at most 200 items of its product per day. The daily profit P, in dollars, can be modeled by the function P(n) = 3.5n - 70 where n is the number of items produced per day and n >= 20 What is the range of the function?
Answer: D
The range of the function is the set of multiples of 3.5 from 28 to 280.
The function P(n) = 3.5n - 70, with n ranging from 20 to 200, yields a minimum profit of 28 dollars when n = 20 and a maximum profit of 280 dollars when n = 200. Thus, the range of the function consists of the multiples of 3.5 within this interval.
A) The set of real numbers
This option is incorrect because the range of the function is not all real numbers. The profit values are specifically determined by the equation and are constrained by the limits on n, resulting in a finite set of outputs.
B) The set of integers greater than 0
This option is incorrect as the range of the function does not include all integers greater than 0. The profit values are defined by the specific outputs of the function, which are limited to a defined interval, not all integers.
C) The set of integers from 20 to 200
This option is incorrect because while n is confined between 20 and 200, the profits generated from this range do not correspond directly to integers in that set. The profit values calculated from the function are not simply integers from this range but rather specific values based on the profit formula.
D) The set of multiples of 3.5 from 28 to 280
This option is correct as the profit function produces values that are multiples of 3.5, starting at 28 when n = 20 and reaching 280 when n = 200. Therefore, the range accurately reflects these multiples within the specified limits.
E) The set of multiples of 3.5 from 0 to 630
This option is incorrect because the range of the profit function does not extend to multiples of 3.5 beyond 280, which is the maximum profit. The range is specifically limited and does not include all multiples up to 630.
Conclusion
The correct answer, D, accurately reflects the range of the profit function as it encompasses the multiples of 3.5 from 28 to 280. All other options either overestimate or misinterpret the defined outputs of the profit function, failing to account for the specific constraints of n in the context of the problem.