# Math 22 Optimization Problems

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## Solving Optimization Sample Problems

**1) Maximum Area**: Find the length and width of a rectangle that has 80 meters perimeter and a maximum area.

Solution: |
---|

Let be the length of the rectangle in meter. |

and be the width of the rectangle in meter. |

Then, the perimeter , so , then |

Area |

, then , so |

Therefore, |

**2) Maximum Volume** A rectangular solid with a square base has a surface area of square centimeters. Find the dimensions that yield the maximum volume.

Solution: |
---|

Let be the length of the one side of the square base in centimeter. |

and be the height of the solid in centimeter. |

Then, the surface area , so |

Volume |

, then , so since is positive. |

Hence, |

Therefore, the dimensions that yield the maximum value is and |

**3) Minimum Dimensions**: A campground owner plans to enclose a rectangular field adjacent to a river. The owner wants the field to contain square meters. No fencing is required along the river. What dimensions will use the least amount of fencing?

Solution: |
---|

Let be the length of two sides that are connected to the river. |

and be the length of the sides that is opposite the river. |

Then, the area , so |

The fence |

, then , so since is positive. Then, |

Therefore, the dimensions of the fence is meters and meters. |

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