Let's go through the solution step-by-step for each part of the problem.
### Given:
The total cost function for producing [tex]\( x \)[/tex] food processors is:
[tex]\[ C(x) = 2100 + 70x - 0.4x^2 \][/tex]
### (A) The Exact Cost of Producing the 41st Food Processor
To find the exact cost of producing the 41st food processor, we need to calculate the difference between the total cost of producing 41 food processors and the total cost of producing 40 food processors.
1. Calculate [tex]\( C(41) \)[/tex]:
[tex]\[ C(41) = 2100 + 70(41) - 0.4(41)^2 \][/tex]
[tex]\[ C(41) = 2100 + 2870 - 0.4(1681) \][/tex]
[tex]\[ C(41) = 2100 + 2870 - 672.4 \][/tex]
[tex]\[ C(41) = 4297.6 \][/tex]
2. Calculate [tex]\( C(40) \)[/tex]:
[tex]\[ C(40) = 2100 + 70(40) - 0.4(40)^2 \][/tex]
[tex]\[ C(40) = 2100 + 2800 - 0.4(1600) \][/tex]
[tex]\[ C(40) = 2100 + 2800 - 640 \][/tex]
[tex]\[ C(40) = 4260 \][/tex]
3. Find the exact cost of producing the 41st food processor:
[tex]\[ \text{Exact cost of the 41st food processor} = C(41) - C(40) \][/tex]
[tex]\[ = 4297.6 - 4260 \][/tex]
[tex]\[ = 37.60 \][/tex]
So,
(A) The exact cost of producing the 41st food processor is \[tex]$37.60.
### (B) The Marginal Cost Approximation
The marginal cost is the derivative of the total cost function with respect to \( x \), which gives us the rate of change of the cost function.
1. Find the marginal cost \( C'(x) \):
\[ C'(x) = \frac{d}{dx} (2100 + 70x - 0.4x^2) \]
\[ C'(x) = 70 - 0.8x \]
2. Evaluate the marginal cost at \( x = 41 \):
\[ C'(41) = 70 - 0.8(41) \]
\[ C'(41) = 70 - 32.8 \]
\[ C'(41) = 37.2 \]
So,
(B) Using the marginal cost, the approximate cost of producing the 41st food processor is \$[/tex]37.20.
