No other 3D model contains more coins than you can accomplish here!
Create the 3D model of a boat or a floating object that transports the largest number of coins in local currency.
HOW TO PROCEED?
To optimize the print model is necessary to calculate how many coins can be placed within an empty 3D model, which uses all the available volume and floats without sinking.
I REMEMBER OF AN ANCIENT GREEK MATHEMATICIAN...
To calculate how much money we can put into our 3D model will take advantage of the Archimedes' Principle; this tells us that:
Any object, wholly or partially immersed in a fluid, is buoyed up by a force equal to the weight of the fluid displaced by the object.
we can write this as:
Archimedes' Force = Weight of Displaced Fluid
In addition, the laws of classical physics say that the weight of an object is calculated as:
Weight = Mass x g = Volume x Density x g
NB: "g" is the acceleration of gravity
HOW DO NOT SINK?
This weight's force of the displaced fluid, is directed from the bottom upwards and is that which allows a body immersed, for example in water, to float.
But sometimes it happens that a body sinks instead of floating; why? It happens when its weight, which is the weight's force that pushes it downwards from above, is greater than the thrust of the weight's force of the displaced fluid.
For our goal will be enough that the weight of our 3D model with its coin does not exceed the weight of the fluid displaced by its immersed volume.
In addition, more fluid volume displaced, heavier can be our model 3D full of coins.
To do this:
- We want the maximum available volume of printing is 100% occupied by our 3D model.
- We want our 3d model is fully immersed in the fluid without sinking.
OK, WE NEED TO MAKE IT FLOAT!
So, to make sure that our 3D model completely immersed can float in the water, its weight must be equal to the Archimedes' force that corresponds to the weight of the displaced fluid:
Weight of 3D Model = Volume of Displaced Fluid x Density of Fluid x g
But the weight of the 3D model is composed by the weight of the filament needed to print all its parts plus the weight of the transported coins:
Weight of Filament + Weight of Coins = Volume of Displaced fluid x Density of Fluid x g
But, as we have already seen, weight = volume x density, so we write:
(Volume of Filament x Density of Filament x g) + (Volume of Coins x Density of Coins x g) = Volume of Displaced Fluid x Density of Fluid x g
IT IS BETTER WITHOUT OR WITH A COVER?
Consider that, for both solutions, you still need to print the base and the vertical walls and the above equation tells us that more filament we use, less money we can charge.
- Without the cover: if we make a 3D model without an upper top, we must leave a minimum of edge above the water surface. To be sure that the coins will not move, we must also create the seats for they. This will avoid a displacement of the ideal center of gravity of the mass and the consequent filling of fluid that could sink the 3D model. In this case will be necessary more filament to create the compartments for the coins. So to maintain the upper edge higher than the surface of the water, we can load less coins.
- With the cover: If we opt for a 3D model with a waterproof top, we can give up the concept of higher edge and inner seats for the benefit of the quantity of coins loaded, but still taking into account the weight of the cover. In addition the top must be printed alone, because the necessary supports for printing all-in-one will make unusable the interior space as cargo bay.
A PAIR OF SIMPLIFICATIONS YET...
We can simplify the last equation in according to the properties of the invariantiva (sorry, I don't know the english word...) equalities, eliminating the acceleration of gravity (g) that always appears:
(Volume of Filament x Density of Filament) + (Volume of Coins x Density of Coins) = Volume of Displaced Fluid x Density of Fluid
In addition, we have already said that we want the volume of the displaced fluid must match the volume of our print area, so:
(Volume of Filament x Density of Filament) + (Volume of Coins x Density of Coins) = Volume of Printer x Density of Fluid
BUT WHICH SHAPE FOR 3D MODEL?
As we can see the equation is independent of the shape of the object into the fluid; It depends only on the volume and the density.
So, would be useless and dispersive try to print our 3D model with a shape different from the print area, that instead we want use at maximum.
This supports us in the choice that we are carrying out to have a 3D model which benefits from the maximum load in the hold at the disadvantage of the hydrodynamics.
They are known or retrievable:
- Volume of Filament: is the volume of the outer perimeter of our 3D model for the thickness of the walls; if with cover we can calculate its volume; if with the inner seats for coins, we can calculate the average density of a single coin plus the filament necessary for its compartment and then we can use this average to calculate the number of coins in the last formula below.
- Density of Filament: we can find it on Wikipedia.
- Volume of Coins: unknown; is the value to be calculated.
- Density of Coins: on Wikipedia if we not find the density, we can find the weight and size of the coins from which to derive the density (weight / volume) .
- Volume Printer: we know it.
- Density Fluid: we can find it on Wikipedia.
So the only unknown factor in this equation is the volume of coins, so we can write:
Volume of Coins = ((Volume of Printer x Density of Fluid) - (Volume of Filament x Density of Filament)) / Density of Coins
Calculated the value of the volume of all the coins, we can divide it by the volume of a single coin and so we obtain the number of coins that our 3D model can carry without sinking:
Number of Coins = Volume of Coins / Volume of one Coin
Thanks to you, Archimede...