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.
A) Determine the total cost at a production level of 50 pizzas a day
B) Determine the production level that would minimize the average daily cost per pizza
C) What is the minimum average daily cost?
http://www.jiskha.com/display.cgi?id=1340080795
To determine the total cost at a production level of 50 pizzas a day (A), we need to substitute x = 50 into the cost function c(x).
So, plugging in x = 50 into the cost function:
c(50) = 0.00025(50)^2 + 8(50) + 10/50
Calculate:
c(50) = 0.00025 * 2500 + 400 + 0.2
= 0.625 + 400 + 0.2
= 400.825
Therefore, the total cost at a production level of 50 pizzas a day is $400.825.
Next, let's determine the production level that would minimize the average daily cost per pizza (B).
To find the production level that minimizes the average daily cost per pizza, we need to find the value of x that minimizes the cost function c(x).
To do this, we can take the derivative of the cost function with respect to x, set it equal to zero, and solve for x.
c'(x) = -0.0005x + 8 - 10/x^2
Setting c'(x) equal to zero:
0 = -0.0005x + 8 - 10/x^2
Multiply both sides by x^2 to eliminate the fraction:
0 = -0.0005x^3 + 8x^2 - 10
Now, we can solve this cubic equation. Unfortunately, solving cubic equations can be complex and time-consuming. So, using numerical methods or graphing software would be more practical to find the approximate value of x that minimizes the average daily cost per pizza.
Finally, to determine the minimum average daily cost (C), substitute the value of x found in step B into the cost function c(x).
Plug the value of x found in step B into the cost function:
c(x) = 0.00025x^2 + 8x + 10/x
Evaluate the cost function at the value of x:
c(x) = 0.00025(estimated value)^2 + 8(estimated value) + 10/(estimated value)
Calculate the minimum average daily cost.