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()=(−2.5). Does the function ℎ have at the same maximum value as ? 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 ()=()−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 parabola represented by the function m(x). The x-coordinate of the vertex is given by -b/2a, where a = -300 and b = 978.

So, x = -978/(2*(-300)) = 1.63 (rounded to two decimal places).

Substituting x = 1.63 into the function m(x), we get:

m(1.63) = -300(1.63)^2 + 978(1.63) - 570 = 386.13

Therefore, the maximum weekly profit is $386.13, and the price of a cup of coffee that produces that maximum profit is $1.63.

B) The function h(x) = -2.5x is a linear function, which means it does not have a maximum value like the quadratic function m(x). Therefore, it cannot have 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 gives the maximum value of the function h(x) = -2.5x. Since this is a linear function, the maximum value is at one of the endpoints of the domain.

Assuming that the domain of the function is the set of non-negative real numbers (since the price of a cup of coffee cannot be negative), the endpoints of the domain are x = 0 and x = infinity.

At x = 0, the function value is h(0) = 0, which is not the maximum value. As x increases towards infinity, the function value decreases without bound, so the maximum value must occur at x = infinity.

However, in the real world, there is usually a limit on how much a coffee house can charge for a cup of coffee, so we need to choose a reasonable upper limit for the domain. Let's say that the maximum price for a cup of coffee is $10.

At x = 10, the function value is h(10) = -2.5(10) = -25, which is the maximum value for the function. Therefore, to produce a maximum profit, Delish Coffee should charge $10 per cup of coffee.

D) The function g(x) = m(x) - 115 represents the weekly profit after a fixed cost of $115 is subtracted. The fixed cost is a cost that the coffee house incurs regardless of how many cups of coffee it sells, such as rent or salaries.

The maximum value of the function g(x) represents the maximum profit after the fixed cost is subtracted. The maximum value of g(x) will be less than the maximum value of m(x) by the amount of the fixed cost.

For example, if the maximum value of m(x) is $500, then the maximum value of g(x) will be $500 - $115 = $385. This means that the coffee house can expect to make a maximum profit of $385 per week after paying the fixed cost.