Certainly! Let's break down each part of the question step-by-step:
### Given Cost Function
[tex]\[ C(x) = 67600 + 400x + x^2 \][/tex]
### a) The cost at the production level 1400
To find the cost at a production level of [tex]\( x = 1400 \)[/tex]:
[tex]\[ C(1400) = 67600 + 400 \cdot 1400 + 1400^2 \][/tex]
The calculated cost at the production level 1400 is:
[tex]\[ C(1400) = 2587600 \][/tex]
### b) The average cost at the production level 1400
The average cost is given by the total cost divided by the number of units produced:
[tex]\[ \text{Average Cost} = \frac{C(x)}{x} \][/tex]
At [tex]\( x = 1400 \)[/tex], the average cost is:
[tex]\[ \text{Average Cost at 1400} = \frac{2587600}{1400} = \frac{12938}{7} \][/tex]
### c) The marginal cost at the production level 1400
The marginal cost is the derivative of [tex]\( C(x) \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[ C'(x) = \frac{d}{dx} (67600 + 400x + x^2) = 400 + 2x \][/tex]
At [tex]\( x = 1400 \)[/tex], the marginal cost is:
[tex]\[ C'(1400) = 400 + 2 \cdot 1400 = 3200 \][/tex]
### d) The production level that will minimize the average cost
First, express the average cost function:
[tex]\[ \text{Average Cost} = \frac{C(x)}{x} = \frac{67600 + 400x + x^2}{x} = \frac{67600}{x} + 400 + x \][/tex]
To find the minimum average cost, take the derivative of the average cost function and set it to zero:
[tex]\[ \frac{d}{dx} \left( \frac{67600}{x} + 400 + x \right) = -\frac{67600}{x^2} + 1 = 0 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ -\frac{67600}{x^2} + 1 = 0 \][/tex]
[tex]\[ \frac{67600}{x^2} = 1 \][/tex]
[tex]\[ x^2 = 67600 \][/tex]
[tex]\[ x = \sqrt{67600} \approx -260 \][/tex]
The production level that minimizes the average cost is:
[tex]\[ x = -260 \][/tex]
### e) The minimal average cost
Substitute [tex]\( x = -260 \)[/tex] into the average cost function:
[tex]\[ \text{Minimal Average Cost} = \frac{67600}{-260} + 400 - 260 = -120 \][/tex]
### Summary of Results
a) Cost at production level 1400: [tex]\( 2587600 \)[/tex]
b) Average cost at production level 1400: [tex]\( \frac{12938}{7} \)[/tex]
c) Marginal cost at production level 1400: [tex]\( 3200 \)[/tex]
d) Production level that will minimize the average cost: [tex]\( -260 \)[/tex]
e) Minimal average cost: [tex]\( -120 \)[/tex]
These results provide a complete understanding of the cost behavior based on the given cost function.
