Final answer:
To keep the profit positive, we identify the roots of the quadratic profit function and determine the interval between them. This interval, with realistic, non-negative x-values, represents when the profit is above zero.
Step-by-step explanation:
To find the set of x-values that will keep the profit positive for the profit formula p = − x² + 130x − 3000, we need to determine the values of x for which p is greater than 0. This involves finding the roots of the equation where p = 0 and understanding the behavior of a quadratic function.
First, we set the profit function equal to zero to find the roots:
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- − x² + 130x − 3000 = 0
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- This is a quadratic equation in standard form, where a = − 1, b = 130, and c = − 3000.
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- We apply the quadratic formula: x = −b ± −sqrt(b² − 4ac) / 2a.
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- By solving this equation, we find the two roots, which represent the x-values where the profit is zero.
Since the coefficient of the x² term is negative, the parabola opens downwards, implying that the profit function is positive between the two roots. The roots divide the x-axis into intervals, within which we can test to find where the profit is positive.
After calculating the roots, we establish the interval of x-values between them. Within this interval, the profit p remains positive. It should be noted that x-values must be realistic in the context of the business, meaning that they should be positive and make sense for the number of products or services offered. Therefore, the set of x-values that keep the profit positive is the interval between the roots, excluding any non-sensical or negative x-values.