The perfect pizza parlour estimates the average daily cost per pizza, in dollars, to be c(x) = 0.00025x^2 + 8x + 10/x
Where x is the number of pizzas made in a day.
This function includes both fixed and variable costs. The fixed costs are represented by the constant term of $10, which includes expenses like rent, utilities, and equipment. The variable costs are represented by the quadratic term of 0.00025x^2 and the linear term of 8x, which increase as more pizzas are made.
To find the average daily cost per pizza, we need to divide the total cost by the number of pizzas made. The total cost is given by c(x), so we can write:
Average cost = c(x) / x
Substituting c(x), we get:
Average cost = (0.00025x^2 + 8x + 10/x) / x
Simplifying, we get:
Average cost = 0.00025x + 8 + 10/x
This formula gives us the average cost per pizza as a function of the number of pizzas made in a day. As x increases, the variable costs dominate and the average cost per pizza increases. However, there is also a decreasing effect from the inverse relationship with x. Therefore, there is an optimal number of pizzas that can be made to minimize the average cost per pizza. This can be found by taking the derivative of the average cost with respect to x and setting it equal to zero.
To estimate the average daily cost per pizza, you are given the cost function:
c(x) = 0.00025x^2 + 8x + 10/x
To find the average daily cost per pizza, we need to find the derivative of the cost function and set it equal to zero to find any critical points.
1. Find the derivative of c(x) using the power rule and the quotient rule:
c'(x) = 0.0005x - 10/x^2 - 10/x^3
2. Set c'(x) equal to zero and solve for x:
0.0005x - 10/x^2 - 10/x^3 = 0
To solve this equation, we need to multiply through by x^3 to eliminate the denominator:
0.0005x^4 - 10x - 10 = 0
This equation is a quartic equation, which can be quite complicated to solve. However, you can use numerical methods or approximation techniques to find the approximate values of x that make the equation true.
Once you find these critical values, plug them back into the cost function c(x) to find the corresponding average daily cost per pizza.
Please note that without additional information, such as the range of possible values for x, it may not be possible to estimate the average daily cost per pizza accurately.