The following function represents the profit P(n), in dollars, that a concert promoter makes by selling tickets for n dollars ea ch: P(n) = -250n2 + 2,500n - 5,250
Part A: What are the zeroes of the above function and what do they represent? Show your work. (4 points)
Part B: Find the maximum profit by completing the square of the function P(n). Show the steps of your work. (4 points)
Part C: What is the axis of symmetry of the function P(n)? (2 points)
1 answer:
A) zeroes P(n) = -250 n^2 + 2500n - 5250 Extract common factor: P(n)= -250 (n^2 - 10n + 21) Factor (find two numbers that sum -10 and its product is 21) P(n) = -250(n - 3)(n - 7) Zeroes ==> n - 3 = 0 or n -7 = 0 Then n = 3 and n = 7 are the zeros. They rerpesent that if the promoter sells tickets at 3 or 7 dollars the profit is zero. B) Maximum profit Completion of squares n^2 - 10n + 21 = n^2 - 10n + 25 - 4 = (n^2 - 10n+ 25) - 4 = (n - 5)^2 - 4 P(n) = - 250[(n-5)^2 -4] = -250(n-5)^2 + 1000 Maximum ==> - 250 (n - 5)^2 = 0 ==> n = 5 and P(5) = 1000 Maximum profit =1000 at n = 5 C) Axis of symmetry Vertex = (h,k) when the equation is in the form A(n-h)^2 + k Comparing A(n-h)^2 + k with - 250(n - 5)^2 + 1000 Vertex = (5, 1000) and the symmetry axis is n = 5.
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