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Bikes produced per day		Total costMarginal cost

1		\$80

2		\$97

3		\$110

4		\$130

5		\$160

6		\$210

7		\$270






Using this table, calculate the marginal cost of each of these quantities of bikes.

The first bike: \$ [ ]

The fourth bike: \$ [ ]

The sixth bike: \$ [ ]

The seventh bike: \$ [ ]

Verified Answer :

  • Define the marginal cost of the \(n\)-th unit as the change in total cost: \(MC(n) = TC(n) - TC(n-1)\).
  • Calculate the marginal cost for the 1st bike: \(TC(1) - TC(0) = \$80 - \$0 = \$80\).
  • Determine the marginal costs for subsequent units by subtracting the previous total cost from the current one:
  • 4th bike: \(TC(4) - TC(3) = \$130 - \$110 = \$20\).
  • 6th bike: \(TC(6) - TC(5) = \$210 - \$160 = \$50\).
  • 7th bike: \(TC(7) - TC(6) = \$270 - \$210 = \$60\).
  • Final answers: The first bike: \(\boxed{\$80}\); the fourth bike: \(\boxed{\$20}\); the sixth bike: \(\boxed{\$50}\); the seventh bike: \(\boxed{\$60}\).

Explanation

  1. Analysis and Formula for Marginal Cost

To determine the marginal cost, we first analyze the relationship between the quantity produced and the total cost. The total cost \(TC(n)\) is given for each quantity \(n\) of bikes. The marginal cost \(MC(n)\) of producing the \(n\)-th bike is defined as the change in total cost resulting from producing that additional unit:
\[MC(n) = TC(n) - TC(n-1)\]
For the first bike (\(n=1\)), we calculate the marginal cost as the cost of producing that unit starting from zero, assuming \(TC(0) = \$0\):
\[MC(1) = TC(1) - TC(0) = \$80 - \$0 = \$80\]

  1. Calculation of Marginal Costs

We now apply the marginal cost formula \(MC(n) = TC(n) - TC(n-1)\) to calculate the costs for the specific quantities requested:

  • For the 4th bike:

\[MC(4) = TC(4) - TC(3) = \$130 - \$110 = \$20\]

  • For the 6th bike:

\[MC(6) = TC(6) - TC(5) = \$210 - \$160 = \$50\]

  • For the 7th bike:

\[MC(7) = TC(7) - TC(6) = \$270 - \$210 = \$60\]

  1. Final Answer Summary

The marginal costs for the requested quantities are:

  • The first bike: \(\$80\)
  • The fourth bike: \(\$20\)
  • The sixth bike: \(\$50\)
  • The seventh bike: \(\$60\)

Examples

In business management, understanding marginal cost is essential for optimizing production levels. For example, if a bicycle shop manager sees that the marginal cost of producing the 7th bike (\(\$60\)) is significantly higher than that of the 4th bike (\(\$20\)), they might decide that expanding production beyond a certain point is not cost-effective. By analyzing these incremental costs, businesses can make informed decisions to maximize profitability and manage their resources efficiently.

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