Finding Maximum and Minimum Values

Congruent squares are cut from the corners of a 1m square piece of tin, and the edges are then turned up to make an open rectangular box. How large should the squares cut from the corners be in order to maximize the volume of the box?

0.1 0.1 0.2 0.3 0.4 0.5 0.6x

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Answer – squares should be 1/6 m

A manufacturer produces cardboard boxes with square bases. The top of each box is a double flap that opens as shown. The bottom of the box has a double layer of cardboard for strength. If each box must have a volume of , what dimensions will minimize the amount of cardboard used?

312 ft

12 1 2 3 4 5 6 7 8x

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400fx

Answer: 2x2x3 feet

A manufacturer wants to design an open box having a square base and a surface area of 108 square inches. What dimensions will produce a box with a maximum volume?

12 1 2 3 4 5 6 7 8 9 10 11 12x

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120Vx

Answer: 6x6x3 inches

A rectangular page is to contain 24 square inches of print. The margins at the top and bottom of the page are to be 1.5 inches and the margins on the left and right are to be 1 inch. What should the dimensions of the page be so that the least amount of paper is used?

246810121416 2 4 6 8 10 12 14 16x

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80fx

Answer – page is 6x9 inches

Four feet of wire is to be used to form a square and a circle. How much of the wire should be used for the square and how much should be used for the circle to enclose the maximum total area?

Answer – Maximum when x = 0, when all the wire is used to form the circle.

0.5 0.5 1 1.5x

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Find the equation of the straight line through (2, 3) with gradient m. Find where this line crosses the axes. Find the minimum area of the triangle enclosed between the line and the axes.

Answer –

Minimum area is 12

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24

A cone has height 12 and radius 6. A cylinder is inscribed in the cone. Find the exact maximum volume of the inscribed cylinder.

12 1 2 3 4 5 6 7 85

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Answer – 64

A cone is inscribed in a sphere with radius 3. Find the exact maximum volume of the cone.

Answer – 323

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A box company produces a box with a square base and no top that has a volume of . Material for the bottom costs and the material for the sides costs . Find the dimensions of the box that minimizes the cost.

38 ft 2$6 / ft2$3/ ft

Answer: 2x2x2 feet

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Mr. Galaty wants to get to the bus stop as quickly as possible. The bus stop is across a grassy park, 2000 meters West and 600 meters North of his starting position. He can walk West along the edge of the park on the sidewalk at a speed of 6 m/sec. He can also travel through the park but only at a rate of 4 m/sec (the park is a favorite place to walk dogs, so he must walk with care). Using your calculator, what path will get him to the bus stop the quickest?

400 400 800 1200 1600 2000 2400 2800

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Answer: When x is about 1463 meters