the total cost (in dollars) of producing x food processors is c(x)=2500 + 30x - 0.4x^{2}. (a) find the exact…

the total cost (in dollars) of producing x food processors is c(x)=2500 + 30x - 0.4x^{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 $

the total cost (in dollars) of producing x food processors is c(x)=2500 + 30x - 0.4x^{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 $

Answer

Explanation:

Step1: Find cost of producing 31 processors

$C(31)=2500 + 30\times31-0.4\times31^{2}$ $C(31)=2500 + 930-0.4\times961$ $C(31)=2500 + 930 - 384.4$ $C(31)=3045.6$

Step2: Find cost of producing 30 processors

$C(30)=2500+30\times30 - 0.4\times30^{2}$ $C(30)=2500 + 900-0.4\times900$ $C(30)=2500 + 900 - 360$ $C(30)=3040$

Step3: Calculate exact cost of 31st processor

The exact cost of the 31st processor is $C(31)-C(30)$. $C(31)-C(30)=3045.6 - 3040=5.6$

Step4: Find marginal cost function

The marginal cost function $C'(x)$ is the derivative of $C(x)$. $C(x)=2500 + 30x-0.4x^{2}$, so $C'(x)=30 - 0.8x$

Step5: Approximate cost using marginal cost

Evaluate $C'(x)$ at $x = 30$. $C'(30)=30-0.8\times30$ $C'(30)=30 - 24=6$

Answer:

(A) $5.6$ (B) $6$