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Section 5.4 I can use calculus to solve optimization problems.

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**1. The sum of two nonnegative numbers is 20. Find the numbers**

(a) if the sum of their squares is to be as large as possible. Let the two numbers be represented by x and 20 – x. makes x = 10 a minimum. Maximum must occur at an endpoint. 0 and 20

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**1. The sum of two nonnegative numbers is 20. Find the numbers**

(b) If the product of the square of one number and the cube of the other is to be as large as possible Let the two numbers be represented by x and 20 – x. 12 20 + _ Max at 12, Min at 20 12 and 8

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**1. The sum of two nonnegative numbers is 20. Find the numbers**

(c) if one number plus the square root of the other is as large as possible. Let the two numbers be represented by x and 20 – x. therefore a max

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**A rectangular pen is to be fenced in using two types of**

fencing. Two opposite sides will use heavy duty fencing at $3/ft while the remaining two sides will use standard fencing at $1/ft. What are the dimensions of the rectangular plot of greatest area that can be fenced in at a total cost of $3600? 3x 1y Therefore max The dimensions of a rectangular plot of greatest area are 300 x 900

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3. A rectangular plot is to be bounded on one side by a straight river and enclosed on the other three sides by a fence. With 800 m of fence at your disposal, what is the largest area you can enclose? x y Therefore a max The largest area you can enclose is 80000

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4. An open-top box with a square bottom and rectangular sides is to have a volume of 256 cubic inches. Find the dimensions that require the minimum amount of material. y x therefore a min 8 x 8 x 4

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_ + Therefore max

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Since A ‘ changes from pos to neg at x = 0.860, max of A occurs at x = 0.860

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7. Find the volume of the largest right circular cylinder that can be inscribed in a sphere of radius 5. r 0.5h R h – height of cylinder r – radius of cylinder R – Given radius of sphere Therefore a max

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Optimization Problems Section 4.5. Find the dimensions of the rectangle with maximum area that can be inscribed in a semicircle of radius 10.

Optimization Problems Section 4.5. Find the dimensions of the rectangle with maximum area that can be inscribed in a semicircle of radius 10.

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