Instructions: Use the given information to answer the questions and interpret key features. Use any method of graphing or solving.The roots of the quadratic function describing the relationship between number of products produced and overall profit margin are x=0𝑥=0 and 100100. The vertex is (50,1000)(50,1000).The maximum profit of $$ dollars is reached when items are produced.The first root tells us that the profit will be 00 when products are produced. The second root says once items are made, the company is no longer making any profit. (They do not have production capacity and have to outsource for anything over 5050.)
Question
Instructions: Use the given information to answer the questions and interpret key features. Use any method of graphing or solving.The roots of the quadratic function describing the relationship between number of products produced and overall profit margin are x=0𝑥=0 and 100100. The vertex is (50,1000)(50,1000).The maximum profit of dollars is reached when items are produced.The first root tells us that the profit will be 00 when products are produced. The second root says once items are made, the company is no longer making any profit. (They do not have production capacity and have to outsource for anything over 5050.)
Solution
The given information describes a quadratic function, which is a parabola. The roots of the function are x=0 and x=100. These are the x-values where the function crosses the x-axis, meaning the profit is zero.
The vertex of the function is (50,1000). This is the highest point of the parabola, indicating the maximum profit of $1000 is reached when 50 items are produced.
The first root, x=0, tells us that the profit will be $0 when no products are produced, which makes sense as there would be no income without any products.
The second root, x=100, tells us that the profit will again be $0 when 100 items are produced. This suggests that the company has reached its production capacity at this point and will need to outsource for additional production, which will eat into the profit margin, bringing it back down to zero.
So, to maximize profit, the company should aim to produce exactly 50 items. Producing less than this will not fully utilize their capacity, while producing more will require outsourcing and reduce the profit.
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