The function f(x) = –x2 + 60x – 116 models the monthly profit, in dollars, a shop makes for repairing windshields, where x is the 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)

Accepted Solution

Answer:A. We factor the equation to get: -(x-25)^2+361 In the form a(x-h)^2+k, the vertex is (h, k), so the vertex is (25, 361).  This means that the studio makes the most profit from selling 25 memberships, and thus makes 361 dollars.B. The x-intercepts are the values of x for which f(x) is 0.  This equation can be factored as (-a+6)(a-44)=0, with solutions 6 and 44.  Therefore, by selling either 6 memberships or 44 memberships, the studio breaks even, neither making nor losing money.Step-by-step explanation:f(x)=−(x2−60x+900−900)−116f(x)=−(x−30)2+784Maximum Profit=784Vertex is (30, 784)maximum profit is when x wind-shields are repaired.−x2+60x−116=0x2−60x+116=0x2−2x−58x+116=0x(x−2)+58(x−2)=0(x−2)(x−58)=0x=2,58x intercepts are x=2, x=58at these points there is no profit or loss.