When q thousand items of a new product have been sold, the total profits P in th

Question

When q thousand items of a new product have been sold, the total profits P in thousands of dollars is approximated by: P = q(-4q + 25) - (46 + 5q) A) Suppose -22 thousand dollars of profits have been earned. (It is possible to have negative profits in this context. This means the item is not yet profitable.) Find a quadratic equation in the form aq^2 + bq + c = 0 you can solve to determine the number of items sold so far. Don't forget that q is your variable. Answer: The equation is. (Use the variable q.) Preview B) Solve your equation in part [a] by factoring and use your answer to indicate the smallest number of items that need to be sold in order to attain a profit of -22 thousand dollars. Answer: To attain profits of -22 thousand dollars, thousand items must be sold. (And this is the smallest number of these items needed to attain that profit level.)

Explanation / Answer

P = q ( -4q + 25 ) - (46 + 5q )

plugging P = -22

-22 = - 4q^2 + 25q - 46 - 5q

combining like terms

- 22 = - 4q^2 + 20q - 46

adding 22 on both sides

- 4q^2 + 20q - 24 = 0

hence the equation is

- 4q^2 + 20q - 24 = 0

b) solving the equation for q

dividing entire equation by -4

q^2 - 5q + 6 = 0

( q - 2 ) ( q - 3 ) = 0

q = 2 , 3

to attain profit of -22 thousands dollars , 2 thousand items must be sold

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