The function C(x)=300x+6,000 represents the cost to produce x number of items. How many items should be produced so that the average cost is less than $500?

Final answer:

To reduce the average cost per item to less than $500, it is necessary to produce more than 20 items according to the function C(x)=300x+6000.

Explanation:

The function C(x)=300x+6,000 represents the cost to produce x items. The average cost per item is given by C(x)/x. We need to find when this average cost is less than $500.

Setting up the inequality, we get C(x)/x < 500. Substituting the value of C(x) into the inequality, we get (300x + 6000)/x < 500. Simplifying this inequality, we end up with: x > 20.

Therefore, more than 20 items need to be produced for the average cost to be less than $500.

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Answer:

More than 30 items

Step-by-step explanation:

C(x)=300x+6,000

First, find c(x), the average cost function.

c(x)c(x)=C(x)x=300x+6,000x

The average cost function is shown below.

c(x)=300x+6,000x

We want the function c(x) to be less than 500.

c(x)<500

Substitute the rational expression for c(x).

300x+6,000x<500x≠0

Subtract 500 to get 0 on the right.

300x+,6000x−500<0

Find the LCD, and rewrite the left side as one quotient.

300x+6,000x−500xx−200x+6,000x<0<0

Factor the numerator to show all factors.

−200(x−30)x<0

Find the critical points when the numerator or denominator are equal to 0.

−200(x−30)−200≠0x−30x=0x=0=0=30

More than 30 items must be produced to keep the average cost below $500 per item.