Question

the total cost (in dollars) of producing x food processors is $c(x) = 1600 + 20x - 0.3x^2$.(a) find the exact cost of producing the 31st food processor.(b) use the marginal cost to approximate the cost of producing the 31st food processor.(a) the exact cost of producing the 31st food processor is $\square$.(b) using the marginal cost, the approximate cost of producing the 31st food processor is $\square$.

Explanation:

Step1: Calculate the exact cost

The exact cost of producing the \(n\)th unit is \(C(n)-C(n - 1)\).
For \(n = 31\), \(C(31)=1600+20\times31-0.3\times31^{2}\)

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\(C(30)=1600+20\times30-0.3\times30^{2}\)

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\(C(31)-C(30)=1931.7 - 1930=1.7\)

Step2: Calculate the marginal cost

The marginal cost function \(C^\prime(x)\) is the derivative of \(C(x)\).
Since \(C(x)=1600 + 20x-0.3x^{2}\), then \(C^\prime(x)=\frac{d}{dx}(1600)+\frac{d}{dx}(20x)-\frac{d}{dx}(0.3x^{2})\)
Using the power rule \(\frac{d}{dx}(x^{n})=nx^{n - 1}\) and \(\frac{d}{dx}(a)=0\) (where \(a\) is a constant)
\(C^\prime(x)=0 + 20-0.6x\)
When \(x = 30\) (because marginal cost at \(x\) approximates the cost of \((x + 1)\)th unit), \(C^\prime(30)=20-0.6\times30\)

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Answer:

(A) \(1.7\)
(B) \(2\)