Answer :
Certainly! Let's go through each part of the problem step by step.
### Given Cost Function
[tex]\[ C(x) = 12100 + 800x + x^2 \][/tex]
### a) The cost at the production level [tex]\( x = 1650 \)[/tex]:
To find the cost at the production level 1650, substitute [tex]\( x = 1650 \)[/tex] into the cost function:
[tex]\[ C(1650) = 12100 + 800(1650) + (1650)^2 \][/tex]
[tex]\[ C(1650) = 12100 + 1320000 + 2722500 \][/tex]
[tex]\[ C(1650) = 4054600 \][/tex]
Thus, the cost at the production level 1650 is:
[tex]\[ 4,054,600 \][/tex]
### b) The average cost at the production level [tex]\( 1650 \)[/tex]:
The average cost function is given by:
[tex]\[ \text{Average Cost} = \frac{C(x)}{x} \][/tex]
At [tex]\( x = 1650 \)[/tex]:
[tex]\[ \text{Average Cost} = \frac{C(1650)}{1650} \][/tex]
[tex]\[ \text{Average Cost} = \frac{4054600}{1650} \][/tex]
[tex]\[ \text{Average Cost} \approx 2457.33 \][/tex]
Thus, the average cost at the production level 1650 is approximately:
[tex]\[ 2457.33 \][/tex]
### c) The marginal cost at the production level [tex]\( x = 1650 \)[/tex]:
The marginal cost is the derivative of the cost function [tex]\( C(x) \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[ \frac{dC}{dx} = 800 + 2x \][/tex]
At [tex]\( x = 1650 \)[/tex]:
[tex]\[ \text{Marginal Cost} = 800 + 2(1650) \][/tex]
[tex]\[ \text{Marginal Cost} = 800 + 3300 \][/tex]
[tex]\[ \text{Marginal Cost} = 4100 \][/tex]
Thus, the marginal cost at the production level 1650 is:
[tex]\[ 4100 \][/tex]
### d) The production level that will minimize the average cost:
To minimize the average cost, we need to find the critical points of the average cost function:
[tex]\[ \text{Average Cost} = \frac{C(x)}{x} = \frac{12100 + 800x + x^2}{x} = \frac{12100}{x} + 800 + x \][/tex]
Take the derivative of the average cost function with respect to [tex]\( x \)[/tex] and set it to zero to find the critical points:
[tex]\[ \frac{d}{dx} \left( \frac{12100}{x} + 800 + x \right) = -\frac{12100}{x^2} + 1 = 0 \][/tex]
[tex]\[ -\frac{12100}{x^2} + 1 = 0 \][/tex]
[tex]\[ \frac{12100}{x^2} = 1 \][/tex]
[tex]\[ x^2 = 12100 \][/tex]
[tex]\[ x = \sqrt{12100} \][/tex]
[tex]\[ x = 110 \][/tex]
Thus, the production level that will minimize the average cost is:
[tex]\[ 110 \][/tex]
### e) The minimal average cost:
To find the minimal average cost, substitute the production level [tex]\( x = 110 \)[/tex] into the average cost function:
[tex]\[ \text{Minimal Average Cost} = \frac{C(110)}{110} \][/tex]
[tex]\[ C(110) = 12100 + 800(110) + (110)^2 \][/tex]
[tex]\[ C(110) = 12100 + 88000 + 12100 \][/tex]
[tex]\[ C(110) = 112200 \][/tex]
[tex]\[ \text{Minimal Average Cost} = \frac{112200}{110} \][/tex]
[tex]\[ \text{Minimal Average Cost} \approx 1020 \][/tex]
Thus, the minimal average cost is:
[tex]\[ 1020 \][/tex]
To summarize:
a) The cost at the production level 1650 is [tex]\( 4,054,600 \)[/tex].
b) The average cost at the production level 1650 is approximately [tex]\( 2457.33 \)[/tex].
c) The marginal cost at the production level 1650 is [tex]\( 4100 \)[/tex].
d) The production level that will minimize the average cost is [tex]\( 110 \)[/tex].
e) The minimal average cost is [tex]\( 1020 \)[/tex].
### Given Cost Function
[tex]\[ C(x) = 12100 + 800x + x^2 \][/tex]
### a) The cost at the production level [tex]\( x = 1650 \)[/tex]:
To find the cost at the production level 1650, substitute [tex]\( x = 1650 \)[/tex] into the cost function:
[tex]\[ C(1650) = 12100 + 800(1650) + (1650)^2 \][/tex]
[tex]\[ C(1650) = 12100 + 1320000 + 2722500 \][/tex]
[tex]\[ C(1650) = 4054600 \][/tex]
Thus, the cost at the production level 1650 is:
[tex]\[ 4,054,600 \][/tex]
### b) The average cost at the production level [tex]\( 1650 \)[/tex]:
The average cost function is given by:
[tex]\[ \text{Average Cost} = \frac{C(x)}{x} \][/tex]
At [tex]\( x = 1650 \)[/tex]:
[tex]\[ \text{Average Cost} = \frac{C(1650)}{1650} \][/tex]
[tex]\[ \text{Average Cost} = \frac{4054600}{1650} \][/tex]
[tex]\[ \text{Average Cost} \approx 2457.33 \][/tex]
Thus, the average cost at the production level 1650 is approximately:
[tex]\[ 2457.33 \][/tex]
### c) The marginal cost at the production level [tex]\( x = 1650 \)[/tex]:
The marginal cost is the derivative of the cost function [tex]\( C(x) \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[ \frac{dC}{dx} = 800 + 2x \][/tex]
At [tex]\( x = 1650 \)[/tex]:
[tex]\[ \text{Marginal Cost} = 800 + 2(1650) \][/tex]
[tex]\[ \text{Marginal Cost} = 800 + 3300 \][/tex]
[tex]\[ \text{Marginal Cost} = 4100 \][/tex]
Thus, the marginal cost at the production level 1650 is:
[tex]\[ 4100 \][/tex]
### d) The production level that will minimize the average cost:
To minimize the average cost, we need to find the critical points of the average cost function:
[tex]\[ \text{Average Cost} = \frac{C(x)}{x} = \frac{12100 + 800x + x^2}{x} = \frac{12100}{x} + 800 + x \][/tex]
Take the derivative of the average cost function with respect to [tex]\( x \)[/tex] and set it to zero to find the critical points:
[tex]\[ \frac{d}{dx} \left( \frac{12100}{x} + 800 + x \right) = -\frac{12100}{x^2} + 1 = 0 \][/tex]
[tex]\[ -\frac{12100}{x^2} + 1 = 0 \][/tex]
[tex]\[ \frac{12100}{x^2} = 1 \][/tex]
[tex]\[ x^2 = 12100 \][/tex]
[tex]\[ x = \sqrt{12100} \][/tex]
[tex]\[ x = 110 \][/tex]
Thus, the production level that will minimize the average cost is:
[tex]\[ 110 \][/tex]
### e) The minimal average cost:
To find the minimal average cost, substitute the production level [tex]\( x = 110 \)[/tex] into the average cost function:
[tex]\[ \text{Minimal Average Cost} = \frac{C(110)}{110} \][/tex]
[tex]\[ C(110) = 12100 + 800(110) + (110)^2 \][/tex]
[tex]\[ C(110) = 12100 + 88000 + 12100 \][/tex]
[tex]\[ C(110) = 112200 \][/tex]
[tex]\[ \text{Minimal Average Cost} = \frac{112200}{110} \][/tex]
[tex]\[ \text{Minimal Average Cost} \approx 1020 \][/tex]
Thus, the minimal average cost is:
[tex]\[ 1020 \][/tex]
To summarize:
a) The cost at the production level 1650 is [tex]\( 4,054,600 \)[/tex].
b) The average cost at the production level 1650 is approximately [tex]\( 2457.33 \)[/tex].
c) The marginal cost at the production level 1650 is [tex]\( 4100 \)[/tex].
d) The production level that will minimize the average cost is [tex]\( 110 \)[/tex].
e) The minimal average cost is [tex]\( 1020 \)[/tex].
