Question

A coffee house finds that its weekly profit (measured in dollars) as a function of the price (measured in dollars) it charges per cup of coffee is given by the function, defined by m(x)=−300^2+978−570 where P=m(x).

A) Determine the maximum weekly profit and the price of a cup of coffee that produces that maximum profit.


B) The profit function for Delish Coffee (another coffee house) is defined by the function ℎ where ℎ(x)=m(x−2.5). Does the function ℎ have the same maximum value? Explain.


C) What is the price per cup of coffee that Delish Coffee must charge to produce a maximum profit? Explain.


D) Describe the meaning of k(x)=m(x)−115 and then compare/contrast the maximum value for each function.

Answer 1

Explanation:

A) To find the maximum weekly profit, we need to find the vertex of the quadratic function m(x)=-300x^2+978x-570. The x-coordinate of the vertex is -b/2a, where a=-300 and b=978. Therefore, x=-978/(-600) = 1.63. To find the corresponding y-coordinate, we substitute x=1.63 into the function m(x) and get m(1.63)=-300(1.63)^2+978(1.63)-570 = 346.62. So the maximum weekly profit is $346.62, and the price of a cup of coffee that produces that maximum profit is $1.63.

B) The function h(x) = m(x - 2.5) shifts the graph of m(x) 2.5 units to the right. This means that the vertex of h(x) is located at x = 1.63 + 2.5 = 4.13. The value of the vertex of h(x) is the same as the value of the vertex of m(x) since shifting the graph horizontally does not affect the maximum value of the function. Therefore, h(x) has the same maximum value as m(x).

C) To find the price per cup of coffee that Delish Coffee must charge to produce a maximum profit, we need to find the x-value that maximizes the function h(x) = m(x - 2.5). From part A, we know that the maximum value of m(x) occurs at x = 1.63. So the maximum value of h(x) occurs at x = 1.63 + 2.5 = 4.13. Therefore, Delish Coffee must charge $4.13 per cup of coffee to produce a maximum profit.

D) The function k(x) = m(x) - 115 is the profit function for a coffee house that charges $115 less per cup of coffee than the coffee house in part A. The maximum value of k(x) will occur at the same x-value as m(x), since the value of x that maximizes the profit is independent of the constant term (-115 in this case). However, the maximum value of k(x) will be 115 dollars less than the maximum value of m(x), since k(x) is the profit function for a coffee house that charges $115 less per cup of coffee.