Question

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)

Answer 1

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.

Answer 2

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)−116

f(x)=−(x−30)2+784

Maximum Profit=784

Vertex is (30, 784)

maximum profit is when x wind-shields are repaired.

−x2+60x−116=0

x2−60x+116=0

x2−2x−58x+116=0

x(x−2)+58(x−2)=0

(x−2)(x−58)=0

x=2,58

x intercepts are x=2, x=58

at these points there is no profit or loss.