Isosceles triangles are usually represented with the different side as the base, so that their axial symmetry is more evident. But if in our isosceles last week we used one of the two equal sides of the triangle as a base, it is obvious that the one with the greatest area will be the one with the greatest height, that is, the isosceles right angle. Therefore, its third side will measure 10√2 cm = approximately 14.14 cm and the maximum area sought will be 50 cm².
The beer can problem, which has sparked extensive debate among readers (see last week’s comments), at first glance seems unrelated to the triangle problem, and yet they require the same change of perspective, Because in both cases it is advisable to “knock down” the corresponding figure.
Imagine that they tell you that the problem has already been solved and, therefore, the amount of beer for which the center of gravity is as low as possible remains in the can. How can you verify that this is true? Very simple: you put the can (upright) in the freezer and wait for the beer to solidify. Then you balance the can horizontally on a fulcrum and with your Therefore, the center of gravity is the center of said circular surface, and that is its lowest possible point. Because? If we added a little beer, the can would fall towards the empty side, which would weigh a little more, and, therefore, the center of gravity would be higher, and if we removed a little beer, the can would also fall towards the empty side , since now the full side would weigh a little less, then the center of gravity would also rise; Therefore, if both adding and removing beer the center of gravity rises, it means that it is at its lowest point.
Now, with the horizontal can in balance, it is not difficult to calculate the height at which the lowest center of gravity is. Neglecting the weight of the lids (which, because they are at a different distance from the fulcrum, do not affect the balance in the same way), calling a and b, respectively, the lengths of the portions of the empty and full can, V the weight of the can empty and P to the weight of the full can, we have that:
a²V = b²P, from where a/b = √P/√V
And since the can weighs 9 times more full than empty, a/b = 3, that is, the beer occupies 1/4 of the height of the can, and the desired center of gravity is 5 cm from the base.
We are going to verify, applying the law of the lever (the product of the force and the length of your arm is equal on both sides), that the horizontal can, filled to that point with beer, is in balance when resting it on a point that is 5 cm from the base:
The filled part weighs 360/4 + 45/4 = 101.25 g and its center of gravity is 2.5 cm from the fulcrum. The empty part weighs 3 x 45/4 = 33.75 g and is 7.5 cm from the fulcrum, and since 101.25 x 2.5 = 33.75 x 7.5 = 253.125, the can is in equilibrium.
Give me a point of support…
And speaking of the lever, let us remember that Archimedes said, in reference to its force multiplier power: “Give me a point of support and I will lift the world”, because a small force with a long arm can lift a large weight located near the fulcrum. . And there are other ways to multiply force by playing with the conservation of work, which is the product of force times the space traveled: with the help of different gadgets, such as the lever, a small force traveling over a large space can become a large force. that runs through a small space, which can be very useful on some occasions. For example, if your car, while driving down a muddy country road, sinks into the mud and there is no way to get it out no matter how hard you push. It is a very irritating situation, since it would be enough to move it a few inches to free the wheels. But don’t despair: you have a long rope in the trunk and there are trees nearby… What can you do to get out of trouble?
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