Let C(x)=40x-0.05x^2 represent the total cost of producing x baskets. Calculate all of the following:
C(100), C(103), C'(x), C'(100).
Now use those answers to demonstrate that
C(103)≈ C(100) + 3 - C'(100), where ≈ means "is approximately equal to".
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To solve this math problem, we will follow the given steps:
1. Calculating C(100):
C(x) = 40x - 0.05x^2
C(100) = 40(100) - 0.05(100)^2
= 4000 - 0.05(10000)
= 4000 - 500
= 3500
2. Calculating C(103):
C(x) = 40x - 0.05x^2
C(103) = 40(103) - 0.05(103)^2
= 4120 - 0.05(10609)
= 4120 - 530.45
= 3589.55
3. Calculating C'(x):
To find the derivative of C(x), we differentiate the function with respect to x. The derivative of 40x is 40, and the derivative of -0.05x^2 is -0.1x. Therefore:
C'(x) = 40 - 0.1x
4. Calculating C'(100):
C'(x) = 40 - 0.1x
C'(100) = 40 - 0.1(100)
= 40 - 10
= 30
Now using the answers obtained above, we will demonstrate the given equation: C(103) ≈ C(100) + 3 - C'(100)
C(103) = 3589.55
C(100) = 3500
C'(100) = 30
Plugging the values into the equation:
3589.55 ≈ 3500 + 3 - 30
Simplifying the equation:
3589.55 ≈ 3473
The approximate equality statement holds as both sides are similar enough.
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