Question

The marginal cost of producing the xth box of light bulbs is 3 + x²/1,000 dollars. Determine how much is added to the total cost by a change in production from x = 30 to x=90 boxes.

Answer 1

To find the added total cost when increasing production from 30 to 90 boxes, we calculate the integral of the marginal cost function
`3 + x²/1,000` over the interval from 30 to 90, resulting in an added cost of
`$324`.

Step-by-step explanation:

The student is asking about the concept of marginal cost, which is a term used in economics to describe the additional cost of producing one more unit of a good. In this case, the good is boxes of light bulbs, and the equation given is the marginal cost function. To determine the added total cost of increasing production from
`30 to 90 boxes`, we need to calculate the integral of the marginal cost function over this range.

We have the marginal cost function
`MC = 3 + x²/1,000.`

To find the added total cost (ΔTC) when production is increased from
`x = 30 to x = 90`, we integrate the MC function over this interval:

• 
  `∫_(30)^(90) (3 + x²/1,000) dx`

This results in the added total cost from producing box 30 to box 90. The integral of this function is:

• 
  `∫ (3 dx) + ∫ (x²/1,000 dx)`
• 
  `3x |_(30)^(90) + (1/3,000)x³ |_(30)^(90)`
• 
  `3(90) - 3(30) + (1/3,000)(90³) - (1/3,000)(30³)`
• 
  `180 - 90 + (729,000/3,000) - (27,000/3,000)`
• 
  `90 + 243 - 9`
• 
  `ΔTC = 324 dollars`

Therefore, the added total cost by increasing production from
`30 boxes to 90 boxes is $324.`