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Question 5 • Mathematics III

A company plans to sell a new type of vacuum cleaner for 280 dollars each. The company’s financial planner estimates that the cost, y , of manufacturing the vacuum cleaners is a quadratic function with a y -intercept of 11 comma 000 and a vertex of open paren 500 comma 24 comma 000 close paren Which system of equations can be used to determine how many vacuums must be sold for the company to make a profit?

Answer
A
y is equal to 280 x y is equal to 0 point 0 5 2 times open paren x minus 500 close paren squared plus 24 comma 000
B
y is equal to 280 x y is equal to negative 0 point 0 5 2 times open paren x minus 500 close paren squared plus 11 comma 000
C
y is equal to 280 x y is equal to negative 0 point 0 5 2 times open paren x minus 500 close paren squared plus 24 comma 000
D
y is equal to 280 x y is equal to 0 point 0 5 2 x squared plus 11 comma 000

Explanation

Revenue from selling x vacuum cleaners is y = 280x. The cost function has vertex (500, 24,000), so it can be written as y = a(x − 500)² + 24,000. Using the y-intercept (0, 11,000) gives 11,000 = 250,000a + 24,000, so a = −0.052. Thus choice C gives the revenue and cost equations whose intersection identifies the break-even sales amounts.

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