Chapter 5.1, Problem 75E

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10th Edition

Ron Larson

ISBN: 9781305860919

Textbook Problem

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**Economics: Marginal Benefits and Costs** The table gives the marginal benefit and marginal cost of producing *x* units of a product for a given company. (a) Plot the points in each column and use the *regression* feature of a graphing utility to find a linear model for marginal benefit and a quadratic model for marginal cost as functions of the units produced *x*. (b) Use integration to find the benefit *B* and cost *C* equations. Assume *B*(0) = 0 and *C*(0) = 425. (*c*) Find the intervals in which the benefit exceeds the cost of producing *x* units. Make a recommendation for how many units the company should produce based on your findings. *(Source: Adapted from Taylor, Economics, Fifth Edition)*

Number of units | 1 | 2 | 3 | 4 | 5 |

Marginal benefit | 330 | 320 | 290 | 270 | 250 |

Marginal cost | 150 | 120 | 100 | 110 | 120 |

Number of units | 6 | 7 | 8 | 9 | 10 |

Marginal benefit | 230 | 210 | 190 | 170 | 160 |

Marginal cost | 140 | 160 | 190 | 250 | 320 |

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**(a)**

To determine

**To graph:** The marginal benefit, marginal cost and number of units produce by the company and find a linear model for marginal benefit and a quadratic model for marginal cost as a function of the units produce *x*. The tabulated data for number of units, marginal benefit and marginal cost is given as,

Number of units | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

Marginal benefit | 330 | 320 | 290 | 270 | 250 | 230 | 210 | 190 | 170 | 160 |

Marginal cost | 150 | 120 | 100 | 110 | 120 | 140 | 160 | 190 | 250 | 320 |

Explanation

**Given Information:**

The tabulated data for number of units, marginal benefit and marginal cost is given as,

Number of units | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

Marginal benefit | 330 | 320 | 290 | 270 | 250 | 230 | 210 | 190 | 170 | 160 |

Marginal cost | 150 | 120 | 100 | 110 | 120 | 140 | 160 | 190 | 250 | 320 |

**Formula used:**

The sum rule of basic integration

The simple power rule of integration

Use ti-

Step-1: Open ti-

Step-2: Press “

Step-3: Press “STAT” key and press on “

Step-4: Press “STAT” key and go to “CALC”.

Step-5: Choose “

Step-6: Note down the values of Linear Regression model.

Here for this given data Linear Regression model

So, the equation becomes

Step-1: Open ti-

Step-2: Press “

Step-3: Press “STAT” key and press on “

Step-4: Press “STAT” key and go to “CALC”.

Step-5: Choose “

Step-6: Note down the values of Linear Regression model

**(b)**

To determine

**To calculate:** The equation of the benefit *B* and cost *C* to produce *x* units of a product when

Number of units | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

Marginal benefits | 330 | 320 | 290 | 270 | 250 | 230 | 210 | 190 | 170 | 160 |

Marginal cost | 150 | 120 | 100 | 110 | 120 | 140 | 160 | 190 | 250 | 320 |

**(c)**

To determine

**To calculate:** The intervals for which the benefit produces by the company exceed the cost to produce *x* units and also estimate the number of units the company should produce. The data that shows the marginal benefit and marginal cost for the number of unit’s produce is tabulated below.

Number of units | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |

Marginal benefits | 330 | 320 | 290 | 270 | 250 | 230 | 210 | 190 | 170 | 160 |

Marginal cost | 150 | 120 | 100 | 110 | 120 | 140 | 160 | 190 | 250 | 320 |

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P-20TYSCh-5 P-21TYSCh-5 P-22TYS

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