The perfect Pizza Parlour estimates the average daily cost per pizza, in dollars, to be
C(x) = (0.00025x^2 + 8x + 10) / x
Determine the production level that would minimize the average daily cost per pizza.
What is the minimum average daily cost?
10
To determine the production level that would minimize the average daily cost per pizza, we need to find the critical points of the function C(x) and then determine which one corresponds to the minimum average daily cost.
Step 1: Find the derivative of C(x) with respect to x.
C'(x) = [(2)(0.00025x^2 + 8x + 10)(1)x - (0.00025x^2 + 8x + 10)(1)] / x^2
Simplifying this expression, we get:
C'(x) = [0.0005x^2 + 16x + 20 - 0.00025x^2 - 8x - 10] / x^2
= [0.00025x^2 + 8x + 10] / x^2
Step 2: Set the derivative equal to zero and solve for x to find the critical point(s).
0.00025x^2 + 8x + 10 = 0
This equation cannot be solved using standard methods since it does not factor easily. However, we can use the quadratic formula to find the solutions.
The quadratic formula states that if we have an equation of the form ax^2 + bx + c = 0, then the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 0.00025, b = 8, and c = 10. Plugging these values into the quadratic formula, we get:
x = [-(8) ± √((8)^2 - 4(0.00025)(10))] / 2(0.00025)
Simplifying this expression, we get:
x = [-8 ± √(64 - 0.001)] / 0.0005
Step 3: Calculate the values of x using the quadratic formula.
x = [-8 ± √(63.999)] / 0.0005
Now, we have two possible values for x:
x₁ = [-8 + √(63.999)] / 0.0005
x₂ = [-8 - √(63.999)] / 0.0005
Step 4: Determine which value(s) of x correspond to the minimum average daily cost.
To determine which value of x corresponds to the minimum average daily cost, we need to evaluate the second derivative of C(x). If the second derivative is positive, it means that the critical point corresponds to a minimum, whereas if it is negative, it corresponds to a maximum.
Step 5: Find the second derivative of C(x).
C''(x) = d²C(x) / dx² = [d/dx (0.00025x^2 + 8x + 10)] / x^2
C''(x) = [0.0005x + 8] / x^2
Step 6: Evaluate the second derivative at each critical point.
Substituting the values of x₁ and x₂ into the second derivative, we get:
C''(x₁) = [0.0005(-8 + √(63.999)) + 8] / [(-8 + √(63.999))^2]
C''(x₂) = [0.0005(-8 - √(63.999)) + 8] / [(-8 - √(63.999))^2]
Step 7: Determine the value of x that corresponds to the minimum average daily cost.
Compare the values of the second derivative at x₁ and x₂ to determine which one is positive and which one is negative. The value of x that corresponds to a positive second derivative corresponds to the minimum average daily cost.
Finally, substitute the value of x that corresponds to the minimum average daily cost back into C(x) to find the minimum average daily cost.
To determine the production level that would minimize the average daily cost per pizza, we need to find the value of x that minimizes the function C(x).
To start, we should find the derivative of C(x) with respect to x, and set it equal to zero to find the critical points.
Let's find the derivative of C(x):
C'(x) = [d/dx ((0.00025x^2 + 8x + 10) / x)]
To simplify the derivative, let's rearrange the expression first:
C(x) = 0.00025x + 8 + 10/x
Differentiating, we get:
C'(x) = 0.00025 - 10/x^2
Now, let's set C'(x) equal to zero and solve for x:
0.00025 - 10/x^2 = 0
Multiplying both sides by x^2, we have:
0.00025x^2 - 10 = 0
Now, we can solve this quadratic equation for x.
0.00025x^2 - 10 = 0
0.00025x^2 = 10
x^2 = 10 / 0.00025
x^2 = 40,000
x = √40,000
x ≈ 200
So, the production level that would minimize the average daily cost per pizza is approximately 200.
To find the minimum average daily cost, we substitute this value of x into the original cost function, C(x):
C(200) = (0.00025(200)^2 + 8(200) + 10) / 200
C(200) = (0.00025(40,000) + 1,600 + 10) / 200
C(200) = (10 + 1,600 + 10) / 200
C(200) = 1620 / 200
C(200) = 8.10
Therefore, the minimum average daily cost per pizza is approximately $8.10.