Answer :
Answer:
The exact cost of producing the 21st food processor is $38.52.
The marginal cost to approximate the cost of producing the 21st food processor is $38.24
Step-by-step explanation:
Consider the provided function.
[tex]C(x) = 2500 + 50x -0.28x^2[/tex]
(A) Find the exact cost of producing the 21st food processor
The exact cost producing 21st food processor is C(21)-C(20)
Substitute x=21 in above function.
[tex]C(x) = 2500 + 50(21) -0.28(21)^2[/tex]
[tex]C(x) = 2500 +1050 -123.48[/tex]
[tex]C(x) = 3426.52[/tex]
Substitute x=20 in above function.
[tex]C(x) = 2500 + 50(20) -0.28(20)^2[/tex]
[tex]C(x) = 2500 +1000-112[/tex]
[tex]C(x) = 3388[/tex]
The exact cost producing is:
[tex]C(21)-C(20)=3426.52-3388=38.52[/tex]
Hence, the exact cost of producing the 21st food processor is $38.52.
Part (B) Use the marginal cost to approximate the cost of producing the 21st food processor,
To find the marginal cost first differentiate the function with respect to x.
[tex]C(x) = 2500 + 50x -0.28x^2[/tex]
[tex]C'(x) =50-0.56x[/tex]
Now substitute x=21 in above function.
[tex]C'(x) =50-0.56(21)[/tex]
[tex]C'(x) =50-11.76[/tex]
[tex]C'(x) =38.24[/tex]
The marginal cost to approximate the cost of producing the 21st food processor is $38.24
The total cost of producing food processors is a function of the amount of food processors to be produced.
The costs for the given cases are
- The exact cost of producing the 21st food processor is $38.52
- Cost of producing the 21st food processor is approx $38.24
How to calculate the rate of a function?
If a function is differentiable, let of the form [tex]f(x)[/tex]
Then its rate is its first derivative with respect to x.
It tells about the function's output value's change rate with respect to the input.
The given function is [tex]C(x) = 2500 + 50x - 0.28x^2[/tex]
Its rate (with respect to x) is
[tex]C'(x) = 50 - 0.56x[/tex]
Calculating the needed costs:
A) The exact cost of producing the 21st food processor
[tex]C(21) - C(20) = \text{Cost of producing 21st food processor}\\\\2500 + 50(21) - 0.28(21)^2 - 2500 - 50(20) + 0.28(20)^2 =50 - 41 \times 0.28 = 38.52[/tex]
Thus, The exact cost of producing the 21st food processor is $38.52
B) The marginal cost at x = 21 is
[tex]C'(x) = 50 - 0.56x\\C'(21) = 50 - 0.56(21) = 38.24[/tex]
Thus, cost of producing the 21st food processor is approx $38.24
Thus,
The costs for the given cases are
- The exact cost of producing the 21st food processor is $38.52
- Cost of producing the 21st food processor is approx $38.24
Learn more about marginal costs here:
https://brainly.com/question/7781429