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Answer :
Answer:
15
Step-by-step explanation:
To find the minimum amount of items, x, that need to be sold for the manufacturer to make a profit, we can use the quadratic formula
[tex]x = \frac{-b+\sqrt{b^{2}-4ac } }{2a}; x= \frac{-b-\sqrt{b^{2}-4ac } }{2a}[/tex]
[tex]x= \frac{-700+\sqrt{700^{2}-4(a)(-10000) } }{2(1)}; x= \frac{-700-\sqrt{700^{2}-4(a)(-10000) } }{2(1)}[/tex]
[tex]x= \frac{-700+\sqrt{530000} }{2}; x= \frac{-700-\sqrt{530000} }{2}[/tex]
[tex]x=\frac{-700+100\sqrt{53} }{2}; x=\frac{-700-100\sqrt{53} }{2};[/tex]
[tex]x=-350 + 50\sqrt53 = 14.005; x = -350 - 50\sqrt53 = -714.005[/tex]
In context of the problem, we can only rely on the positive value, as the negative value would lead to a loss of profit.
[tex](-714.005)^2+700(-714.005)-10000=-0.360[/tex]
Furthermore, we must round to the nearest whole number, as you cannot make part of an item.
Lastly, if you were to plug in 14 into the equation for Profit, you would still have a negative number (i.e. a negative profit), thus requiring the manufacturer to make no less than 15 items to make a profit:
[tex](14)^2+700(14)-10000=-4\\\\(15)^2+700(15)-10000=725[/tex]
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Final answer:
To make a profit, the manufacturer must sell 350 items each day.
Explanation:
To determine the number of items that must be sold each day in order to make a profit, we need to find the maximum point on the profit function. The profit function is given by P = -x^2 + 700x - 10000. Here, x represents the number of items sold. To find the maximum point, we can use the vertex formula for a quadratic function: x = -b/2a. In this case, a = -1, b = 700. Substituting these values into the formula, we get x = -700/2(-1) = -700/-2 = 350. Therefore, in order to make a profit, the manufacturer must sell 350 items each day.
