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?
The range of the function is the set of multiples of 3.5 from 28 to 280.
The profit function P(n) = 3.5n - 70 can be evaluated at its minimum and maximum production values, n = 20 and n = 200, respectively. By substituting these values into the function, we find the minimum profit is 28 and the maximum profit is 280, both of which are multiples of 3.5.
This option suggests that the range includes all real numbers, which is incorrect. The profit function has defined limits based on the constraints of production (from 20 to 200 items), producing a specific set of outputs rather than all real numbers.
While the profit values are positive within the defined range, this option incorrectly implies that the range includes all positive integers. The actual profit values are specific multiples of 3.5, not just any positive integers.
This choice incorrectly assumes that the range corresponds to the number of items produced rather than the profit generated. The range of the profit function is determined by the specific profit outputs, which are calculated based on the production levels.
This choice correctly identifies the range of the profit function. By plugging in the minimum and maximum values of n, we find that the profit values start at 28 (when n = 20) and end at 280 (when n = 200), and each of these profits is a multiple of 3.5.
This option incorrectly suggests that the range goes up to 630. The maximum profit calculated from the function is 280, not 630, making this option inaccurate.
The range of the profit function P(n) is specifically limited to the profits generated from producing between 20 and 200 items, which results in multiples of 3.5 from 28 to 280. This highlights how the constraints on production directly influence the outputs of the profit function, emphasizing the relevance of the input limits in determining the correct range.
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