For each row I've added the the Out of all the cans in my pantry, the tuna is the most inefficient, and the condensed milk #2, Eagle Brand, is pretty close to the perfect ratio. A Coke can is not "exactly" a Right Circular Cylinder. A Coke can holds 12 fluid ounces or 355 milliliters. Your average soda can has a height of 5 inches and a curcular top and bottomwith a diameter of 2.5 inches. In the past they were manufactured by cutting a rectangle of material and wrapping this around with a single seam weld to which two circular end caps were attached. The original cans may have been manufactured with an arbitrary aspect ratio, but now we've been programmed to search for items of this shape when we shop. Well done guys! Finally substituting back for the valuation of h reveals the result:Below is a graph of how much material (surface area) cans of variable aspect ratio require. It would not stay still on a shelf, making display and storage hard. To do this, we need our old friend Calculus.To work out the optimal size, we keep the volume of the cylinder fixed, then find the relationship between the radius and the height.Recall the simple formula for the Volume, and we can re-arrange this to get an equation for the height:We're trying to minimize the surface area, and here is the equation for this, into which we will substitute the equation for the height:Next we find the first derivative of this and set it to zero to find the turning points in the function:(The second derivative is positive, confirming that this turning point is a minimum). Rather than sharp edges, these have filleted (rounded) edges to reduce stress concentrations and to make them easier to manufacture.Taken to the limit, a cube would be the most efficient of the square cornered containers. I measured a soda can in order to compute an estimate of this volume. A can of soda has a volume of 35.325 cubic inches and a height of 5 inches.
Iâll not prove it here, but a cube is the format that minimizes the surface area to volume ratio of a cuboid.
Cans (especially drinks ones) are designed to be held in the hand. If the diameter were too narrow (or the container too deep), then fingers would not be able to get inside.Aesthetically, a slightly taller can looks nicer. A can's dimensions are 4.83 inches high, 2.13 inches in diameter at the narrowed top and 2.60 inches throughout the rest of a cylindrical can.These sizes arose in the USA in the 1980s and 1990s. There's an obvious maximum diameter that an adult hand can hold, after that, if you want to increase the volume you have to make the container taller.To open a can, a cutting device has to revolve around the rim. (Initially the soda can is You occasionally seen cuboid-like containers (corned-beef, spam and sardines are the first that come to mind). A soup can of the wrong shape might not attract our attention (or bizarrely might be less attractive) to shoppers. We now have a relationship between the volume and the radius for the minimum surface area. One can simply type in the number of ounces, tell the calculator what unit of measurement to convert to, and it will do it for him or her. The volume of a cylinder is: Given: Volume (V) = 23.7 and height (h) = 4.38 inches. height = 4.75 inches diameter = 2.6 inches circumference = 8.1875 inches By assuming the can is approximately a "Right Circular Cylinder", I can use the formula I learned in high school algebra for estimating volume: I estimated that the volume was about 25.219 cubic inches.This estimate is "OFF". The purpose of a food can is to store food. the volume of an cylindrical can with a radius r cm and height h cm is 128000 c^3, show that the surface area of the can is A=2(22/7)r^2 + 246000/r. How could you manufacturer and fill it? There is no reason why it should cost the same.Related to the above, the end caps are circular, whilst the sides are cut from rectangular stock. This, obviously, will be recycled, but the manufacturer has had to pay for processing the raw material into the sheets from which the circular blanks will be cut. In each number, the first digit is the number of whole inches, and the second two digits are the number of sixteenths of an inch.
This corresponds very closely to the condensed soup can ratio. (Ratio of height to diameter)If the goal was to purely maximize the volume of food that could be stored in a container, the result would be a spherical can. The first number is the can's diameter and the second its height. This container is designed to allow fingers to dip in and pick up nuts to eat.
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