Minimum Surface Area of a Can

A Solve the optimization problem using Excel. In mathematics a minimal surface is a surface that locally minimizes its area.

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A 2 π r 2 2 π r h.

. The volume of the cylinder is constrained to be 355 ml. Soda can needs to remain 173 in3 but we want to minimize the surface area by adjusting the dimensions of radius and height. Fixed we can apply the volume of the can solve for h pug this into the surface area formula and then use the AM GM rule to find the minimum surface area.

The can is NECKED IN at the top ie it is not a perfect cylinder and volume 330 ml the radius of the top is R and fixed as 29mm the slant angle is 70 degrees from horizontal ie it bends in 20degress the slant height is X the height of the perfect cylindrical part of the can is h and the radius of the bottom is r. V 1 0 0 0 c m 3 π r 2 h. The total surface area is calculated as follows.

Surface area of cylinder. D d r r 2 128 r 0 solve for r. Finding the Ratio of Height to Radius.

This is equivalent to having zero mean curvature. The total surface area is minimum if l w h that is if it is a cube. Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame.

Visual in the figure below. The surface area is the areas of all the parts needed to cover the can. SA 4πr 2 2πrh.

The term minimal surface is used because these surfaces originally arose as surfaces that minimized total surface area subject to some constraint. A fun experiment is to dip a loop of wire into a soap solution and pull it out. The answer is 96 π.

If we divide the volume of each can by the surface area of each can this will tell us the ratio of these values but it will not tell us what the optimal shape is for the volume of food held. Consider the problem of designing a cylinder soda can with minimum surface area. Trying to predict what shape the soap film will be often yields a surprise.

You can find the area of the top or the bottom. The surface area formula for a cylinder is π x diameter x diameter 2 height where diameter 2 is the radius of the base d 2 x r so another way to write it is π x radius x 2 x radius height. Cylinder_surface_minpy calculate the height to radius ratio of a minimum surface cylindrical container can r radius circle_area pi r2 circle_circumference 2 pi r h height cylinder_volume pi r2 h surface area of a closed cylinder.

The answer I get is A 3000 500 π 1 3 which is correct because I graphed it using a graphing calculator and both give 553581. R 3745 and S 2644 The cans height can be found from 2 330 r h S. The radius that will result in the minimum surface area is approximately 34365 centimeters and the corresponding height is Y 2 34365 6873 centimenters.

Solve for the derivative of zero. Volume of Cylinder V π r 2 h 128 π eq1 Surface Area of Cylinder S A 2 π r 2 r h eq2 Substitute eq 1 and eq 2. Lateral Surface Area π rs π rr 2 h 2 Base Surface Area π r 2.

Note that the surface area of the bases of the cylinder is not included since it does not comprise part of the surface area of a capsule. Explain the height column of the spreadsheet is a formula for. The minimum surface area of a cylindrical can that holds 255 cubic centimeters is approximately 22261 square centimeters.

Displaystyle A 2 pi r2 2pi r h A 2πr2 2πrh. The surface area of a capsule can be determined by combining the surface area equations for a sphere and the lateral surface area of a cylinder. Substituting r 3745 gives h 7490 The can with the minimum surface area has radius 374 cm and height.

Total surface area 2 lw wh hl Now for a given volume of the parallelepiped. As if by magic a thin transparent film of soap will form across the wire. Recall the formula for volume is V πr2h Then h V πr2 1 Now the surface area of the can is given by SA 2πr2 2πrh 2 So plugging 1 into 2 gives.

Thats the top the bottom and the paper label that wraps around the middle. The area of the top pi r2 the area of the bottom pi r2 the area of the side 2 pi r h or cylinder_surface. The solution involves a sign analysis of the f.

R 500 π 1 3 into A 2 π r 2 2000 r to find the minimum surface area of the 1000ml can. Use the slider to adjust the shape of the cylinder and. We get r 4.

That is the problem is to find the dimensions of a cylinder with a given volume that minimizes the surface area. The surface area of a cylinder is area of two of the circular end caps plus the rectangle that wraps around the edge forming the sides. More accurate values can be obtained by using smaller increments in the value of r in Excel or by zooming in further on your graphic calculator.

Try this to obtain the values. Capsule Surface Area Volume π r 2 43r a Surface Area 2 π r2r a Circular Cone Surface Area Volume 13 π r 2 h. SA 2πr2 2πr V πr2.

A common optimization problem faced by calculus students soon after learning about the derivative is to determine the dimensions of the twelve ounce can that can be made with the least material. Im just getting stuck with the algebra. Displaystyle V 1000cm3 pi r2h V 1000cm3 πr2h.

What are the dimensions of a 12 ounce can that minimize the amount of aluminum required for the top and sides. Surface Area 2Area of top perimeter of top height Surface Area 2pi r 2 2 pi r h In words the easiest way is to think of a can. Minimum surface area of a pepsi can.

Minimum Surface Area Nearly everybody has at some point been fascinated by soap bubbles. What makes the soap bubble take the shape of a perfect sphere and. Volume of cylinder.

So in this case l w h 371 1 3 cm 333522 cm. Describe the formulae you construct in Excel and attach a. Surface area of a cylinder.

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