Calculus I More Max/Min Problems Practice |

This practice exercise covers

**Solving Optimization Word Problems**

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Make a diagram for questions that need one.

**QUESTIONS**

1) The sum of two nonnegative numbers is 10. Find these numbers if:

- a) their product is a minimum.

- b) the sum of their squares is a minimum.

- c) the sum of their squares is a maximum.

2) The sum of two nonnegative numbers is 36. Find these numbers if

the first number plus the square of the second is:

- a) a maximum.

- b) a minimum.

3) A rectangular play yard is being built along the side of a house with fence

on 3 sides and the wall of the house as the 4th side of the yard. If there are

20 metres of fence, find the dimensions that maximize the area of the yard.

4) A farmer with 120 metres of fencing wants to enclose a rectagular pig pen.

The pen will have an internal fence running parallel to one side dividing the area

into two sections, one has twice the length of the other. Find the dimensions to

maximize the area.

5) An 20 cm. square piece of cardboard will be used to create an open box by

cutting equal squares from the corners and folding up the sides.

What dimensions will maximize the volume of such a box?

6) Find the dimensions of the largest rectangle that can be inscribed in a

triangle with sides 6, 8 and 10 centimetres.

7) Find the dimensions of the right triangle with hypotenuse of 2 cm.

with a maximum area.

8) Find the dimensions of the largest rectangle that can be inscribed in a

- a) semi-circle with radius 8 cm.

- b) a circle with radius 4 cm.

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**SOLUTIONS**

1) Let x = 1st part, so 10 – x = 2nd parta) f (x) = x (10 – x) -- a parabola, opens down with a max minimum happens at either endpoint x = 0, or x = 10b) minimize c) maximize |
2) Let x = 1st part, so 36 – x = 2nd parta) f (x) = x + (36 – x)² a parabola that opens up with a minimum maximum happens at the initial endpoint x = 0since we square the 2nd part, we want it big. first part = 0, so second part = 36b) the minimum happens at |

3) |
4) |

5) |
6) with similar triangles, find
Now So |

7)
Setting the numerator = 0, we get |
8) a)
Setting the numerator = 0, we get b) Now we have the entire circle with r = 4. |

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