# calculus

7. A cardboard box manufacturer wishes to make open boxes from rectangular pieces of cardboard with dimensions 40 cm by 60 cm by cutting equal squares from the four corners and turning up the sides. Find the length of the side of the cut-out square so that the box has the largest possible volume. Also, find the volume of the box.

1. 👍
2. 👎
3. 👁
1. let each side of the cut-outs be x cm
So the base is 60-2x by 40-2x and the height is x
Volume = x(60-2x)(40-2x)
= 4x^3 - 200x^2 + 2400x
d(volume)/dx = 12x^2 - 400x + 2400
= 0 for a max of volume
3x^2 - 100x + 600 = 0 ----- I divided by 4
x = (100 ± √(2800)/6
= 25.5 or 7.85

but clearly 0 < x < 20 or else the sides make no sense.

so the cut-outs should be 7.85 by 7.85 cm

Plug 7.85 into my volume expression to get the max volume

1. 👍
2. 👎
2. 6. A right circular cylinder is to be designed to hold 750 cm3 of processed milk, and to use a minimal amount of material in its construction. Find the dimensions for the container.

1. 👍
2. 👎
3. A right circular cylinder is to be designed to hold 750 cm3 of processed milk, and to use a minimal amount of material in its construction. Find the dimensions for the container.

1. 👍
2. 👎
4. If you had a device that could record the Earth's population continuously, would you expect the graph of population versus time to be a continuous ( unbroken) curve? Explain what might cause breaks in the curve.

I'm confused. Thanks in advance for any help.

1. 👍
2. 👎
5. Tony has a rectangular piece of cardboard which is 60 cm by 40 cm. By cutting squares from each corner of the cardboard and turning up the sides, can he form an open-topped box which will hold (a) 6656 cubic centimeter? (b) 9000 cubic centimeters? (c) what is the maximum amount of space he can obtain?

1. 👍
2. 👎

## Similar Questions

1. ### calculus

By cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, an open box may be made. If the cardboard is 14 in. long and 6 in. wide, find the dimensions of the box

2. ### math

A manufacturer of open tin boxes wishes to make use of pieces of tin with dimensions 8 in. by 15 in. by cutting equal squares from the four corners and turning up the sides. a. Let x inches be the length of the side pf the square

3. ### math

a 5cm by 5cm square is cut from each corner of a rectangular piece of cardboard.the sides are folded up to make an open box with a maximum volume.if the perimeter of the base is 50cm,what are the dimensions of the box.

4. ### Mathematics

A cylinder of height 12cm and radius 5cm is made of cardboard use the value of 3.1 to calculate the total surface area of the cardboard needed to make a) a closed cylinder. b) a cylinder open at one end

1. ### Calculus 1

A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 18 in. by 30 in. by cutting out equal squares of side x at each corner and then folding up the sides as in the figure. Express the

2. ### Calculus

A box with an open top is to be made from a square piece of cardboard by cutting equal squares from the corners and turning up the sides. If the piece of cardboard measures 12 cm on the side, find the size of the squares that must

3. ### math

a rectangular sheet of cardboard 4m by 2m is used to make an open box by cutting squares of equal size from the four corners and folding up the sides.what size squares should be cut to obtain the largest possible volume?

4. ### Math

A cardboard manufacturer wishes to make open boxes from square pieces of cardboard of side 12 in. by cutting equal squares from the four corners and turning up the sides. Let x inches be the length of the side of the the square to

1. ### Calculus

Squares with sides of length x are cut out of each corner of a rectangular piece of cardboard measuring 3 ft by 4 ft. The resulting piece of cardboard is then folded into a box without a lid. Find the volume of the largest box

2. ### Geometry

On a rectangular piece of cardboard with perimeter 11 inches, three parallel and equally spaced creases are made. The cardboard is then folded along the creases to make a rectangular box with open ends. Letting x represent the

3. ### Geometry

A box of cereal measures 8 inches high and 2 inches deep. If all surfaces are made of cardboard and the total amount of overlapping cardboard in the box is 7 square inches, how much cardboard is used to make the cereal box?

4. ### Geometry/Math

What Formula would I use to calculate how much my cardboard box with a surface area of 1578cm and a volume of 3078 cm would cost if cardboard was \$0.05 per square inch.