Math

Question (A) Find the exact cost of producing the 71st food processor, given C(x)=1900+40x0.2x2C(x)=1900+40x-0.2x^2. (B) Use the marginal cost to approximate the cost of producing the 71st food processor. (A) The exact cost of producing the 71st food processor is $2,662\$2,662. (B) Using the marginal cost, the approximate cost of producing the 71st food processor is $2,660\$2,660.

Studdy Solution

STEP 1

Assumptions
1. The total cost function for producing xx food processors is given by C(x)=1900+40x0.2x2C(x)=1900+40x-0.2x^2.
2. We need to find the exact cost of producing the 71st food processor.
3. We need to use the marginal cost to approximate the cost of producing the 71st food processor.
4. Marginal cost is the derivative of the total cost function with respect to xx.

STEP 2

To find the exact cost of producing the 71st food processor, we need to evaluate the total cost function C(x)C(x) at x=71x=71.
C(71)=1900+40(71)0.2(71)2C(71)=1900+40(71)-0.2(71)^2

STEP 3

First, calculate the linear term 40(71)40(71).
40(71)=284040(71) = 2840

STEP 4

Next, calculate the quadratic term 0.2(71)20.2(71)^2.
0.2(71)2=0.2×5041=1008.20.2(71)^2 = 0.2 \times 5041 = 1008.2

STEP 5

Now, substitute these values into the total cost function to find C(71)C(71).
C(71)=1900+28401008.2C(71)=1900+2840-1008.2

STEP 6

Calculate the exact cost for the 71st food processor.
C(71)=1900+28401008.2=3731.8C(71)=1900+2840-1008.2 = 3731.8
The exact cost of producing the 71st food processor is C(71)=$3731.8C(71) = \$3731.8.

STEP 7

To approximate the cost of producing the 71st food processor using the marginal cost, we first need to find the marginal cost function, which is the derivative of the total cost function C(x)C(x) with respect to xx.
C(x)=ddx(1900+40x0.2x2)C'(x) = \frac{d}{dx}(1900+40x-0.2x^2)

STEP 8

Differentiate each term of the total cost function with respect to xx.
C(x)=0+400.4xC'(x) = 0 + 40 - 0.4x

STEP 9

Now, evaluate the marginal cost function C(x)C'(x) at x=70x=70 to approximate the cost of producing the 71st food processor.
C(70)=400.4(70)C'(70) = 40 - 0.4(70)

STEP 10

Calculate the value of C(70)C'(70).
C(70)=400.4×70=4028=12C'(70) = 40 - 0.4 \times 70 = 40 - 28 = 12

STEP 11

The marginal cost at x=70x=70 gives us the approximate additional cost of producing one more food processor after the 70th, which is the 71st.
Approximate cost of 71st processor=C(70)+C(70)Approximate\ cost\ of\ 71st\ processor = C(70) + C'(70)

STEP 12

We already have the marginal cost C(70)=12C'(70) = 12. Now, we need to find C(70)C(70), the cost of producing 70 food processors, to use in our approximation.
C(70)=1900+40(70)0.2(70)2C(70) = 1900 + 40(70) - 0.2(70)^2

STEP 13

Calculate the linear term 40(70)40(70).
40(70)=280040(70) = 2800

STEP 14

Calculate the quadratic term 0.2(70)20.2(70)^2.
0.2(70)2=0.2×4900=9800.2(70)^2 = 0.2 \times 4900 = 980

STEP 15

Now, substitute these values into the total cost function to find C(70)C(70).
C(70)=1900+2800980C(70) = 1900 + 2800 - 980

STEP 16

Calculate the cost for producing 70 food processors.
C(70)=1900+2800980=3720C(70) = 1900 + 2800 - 980 = 3720

STEP 17

Now, use the cost of producing 70 food processors and the marginal cost to approximate the cost of producing the 71st food processor.
Approximate cost of 71st processor=3720+12Approximate\ cost\ of\ 71st\ processor = 3720 + 12

STEP 18

Calculate the approximate cost for the 71st food processor.
Approximate cost of 71st processor=3720+12=3732Approximate\ cost\ of\ 71st\ processor = 3720 + 12 = 3732
The approximate cost of producing the 71st food processor using the marginal cost is C(71)$3732C(71) \approx \$3732.
(A) The exact cost of producing the 71st food processor is $3731.8\$3731.8. (B) Using the marginal cost, the approximate cost of producing the 71st food processor is $3732\$3732.

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