word problem

Question

The demand equation for a microwave is p=140-0.0001x, where p is the unit price (in dollars) of the microwave and x is the number of units produced and sold. The cost equation for the microwave is C=80x +150,000, where C is the total cost (in dollars) and x is the number of units produced. The total profit obtained by producing and selling x units is given by P=R-C=xp-C. You are working in the marketing department of the company that produces this microwave, and you are asked to determine a price p that would yield a profit of $9 million. Is this possible? Explain.

Answer 1

P = xp - C

9,000,000 =   x(140 - 0.0001x) - (80x + 150,000)

9,000,000 = 140x - 0.0001x² - 80x - 150,000

9,000,000 = 60x - 0.0001x² -150,000

0.0001x² - 60x + 9,150,000 = 0

x = [60 +/-√(3600 - 4 * 0.0001 * 9,150,000)]/(2 * .0001)

x = [60 +/-√(3600 - 3660)]/(.0002)

x = [60 +/-√(-60)](.0002)

Because the square root of -60 is imaginary, there is no real solution to the problem.  It is not possible to have a price that will produce a profit of $9 million dollars.  When the price goes low enough to cause enough demand to make the profit, the price per item is too low to offset the cost of production.