The function f(x) = –x2 + 60x – 116 models the monthly profit, in dollars, a shop makes for repairing windshields, where x is th
e number of windshields repaired, and f(x) is the amount of profit. Part A: Determine the vertex. What does this calculation mean in the context of the problem? (5 points) Part B: Determine the x-intercepts. What do these values mean in the context of the problem? (5 points)
The function is: f ( x ) = - x² + 60 x - 116 is in the form: f ( x ) = a x² + b x + c Part A: x = -b / 2 a = - 60 / ( - 2 ) = 30 Then we will charge it into the function: f ( 30 ) = - 30² + 60 · 30 - 116 = - 900 +1,800 - 116 = 784 The vertex is ( 30, 784 ). It means that the maximum profit is $784, when 30 windshields are repaired. Part B: x 1/2 = ( -60 +/- √(3600 - 464 )/ (- 2 ) = ( -60 +/- 56 ) / ( - 2 ) x 1 = 2, x 2 = 58 It means that a shop has to repair at least 2 windshields and not more than 58 windshields for making a profit.
