The cost in millions of dollars of building a threestory hig

The cost, in millions of dollars, of building a three-story high school in New York State was estimated to be

C(x) = 1.9 + 0.14x 0.0001x²   (20 x 400)

where x is the number of thousands of square feet. Suppose that you are contemplating building a for-profit three-story high school and estimate that your total revenue will be $0.21 million dollars per thousand square feet. What is the profit function (in millions of dollars)?

P(x) =

What size school should you build in order to break even? (Round your answer to 3 decimal places.)
_____ thousand ft²

Solution

C (x) = 1.9 + 0.14x - 0.0001x² ( 20 x 400) . The Revenue R (x) = 0.21x . The profit function P (x) = R (x) - C(x) =

0.21x - 1.9 - 0.14x + 0.0001x² =  0.0001x² + 0.07x - 1.9. The break even would occur when P(x) = 0. Then

0.0001x² + 0.07x - 1.9. = 0. On multiplying both the sides by 10000, we have x² + 700x - 19000 = 0 Then x =

[ - 700 ± { (700)² - 4(1) ( -19000) } ] / 2 *1 = [ - 700 ± ( 490000 + 76000) ] / 2 = ( - 700 ± 566000)/ 2 = ( - 700 ± 752.329715)/2 . Now, since x can not be negative, we have x = ( - 700 + 752.329715)/2 = 52.329715 / 2 Thus x = 26.16485748 or, x = 26.165 thousand square feet.