Rates of Growth

Introduction:

If I ride my bike to school it is a 6.5 mile trip.  If I do it roundtrip, it doubles to 13 miles.  This rate of growth is called linear.  When you double the trip it doubles the length, when you triple the trip it triples the length.  The same is true if you measure the force on a spring.  It takes twice as much force to pull a spring twice as far.  This all seems kind of obvious, but it is a different story with other things. 

If I paint a 5x5 foot mural, it takes 25 square feet of paint.  If I double the length of the wall on each side, so that it's 10 by 10 feet, then it takes 100 square feet of paint, four times as much.  For the paint, when I double the wall length I use 4x as much paint, and when I triple the wall length I use 9x as much paint.  When I drop a ball from a cliff, it falls 16 feet in one second, but if I double the time it falls 64 feet (4x as much), and if I triple the time to three seconds it falls 144 feet (8x as much).  This kind of growth rate is called quadratic or n².

To fill up my 5' x 5' x 5' hot tub it takes 125 cubic feet of water.  If I double the dimensions of the tub,  my 10' x 10' x 10' hot tub will hold 1000 cubic feet of water, or 8x times as much.   The rate of growth is cubic.  If you change the input by a factor of k then you change the output by a factor of k³.  If I triple the dimensions, then how many times greater is the volume of water?  Answer:  27 times.

(Optional:  For those of you who are sophisticated algebra users, here is a proof.  Let the cubic relationship be cx³, for some number c > 0.  When the input x is changed by a factor of k, the output c(kx)³ = ck³x³ = k³(cx³), is changed by a factor of k³.

Applications:

1.  A six foot man weighs 180 pounds.  What would you expect a 7 foot man to weigh?  People are mot cube shaped or perfectly proportioned, and there are huge differences in peoples' weight.  Neverthless there are good and bad ways to try to estimate the solution to thisd question.  Is a man a 1-dimensional, 2-dimensional or 3-dimensional object?  Is a man like the distance, the wall, or the pool in terms of dimension?  A man is 3-dimensional.  Therefore we must use the n³ rate of growth.  Seven feet is 7/6 times 6 feet, so you would expect the person's weight to increase by a factor of (7/6)³ = 1.59 approximately.  Therefore we expect the weight of a 7 foot man to be about 180 x 1.59 = 286.2.   I have heard otherwise intelligent people insist that it should be more like 180 x (7/6) = 210. 

2.  When cells grow to a certain size, their rate of growth slows until they stop growing entirely. They have reached their size limit.  At this point, fluid diffuses from the cell and one of these larger cells divides into two smaller cells, and the rate of growth again increases.  Why is this?  Why do you think a cell has a size limit?  Think about what this has to do with rates of growth before you read on.

The volume of the cell is a 3-dimensional quantity, the surface area of the cell is a 2-dimensional quantity, and the diameter of the cell is a one-dimensional quantity.  That means if the diameter of the cell doubles, then the surface area will grow by 4x, and the volume will grow by 8x.   If a cell keeps growing, the pressure per square unit of the fluid inside the cell on the cell membrane will continually increase because the volume of fluid is increasing faster than the surface of the containing membrane.   At some point, the pressure of the inner fluid on the cell membrane will cause the fluid to diffuse out of the cell, and the cell stops growing.  If the membrane and fluid grew at the same rate, there might be no limit to the size of a cell! 

3.  For a uniform material, the weight it can carry is proportional to its cross-sectional area (quadratic). Therefore if you double all the dimensions of a stone building supported on stone pillars, the weight (cubic) increases by a factor of 8, but the supporting capacities only increase by a factor of four.   If you keep making the building larger, there is a limit to how many times the dimensions can be doubled before the building will crumble. 

This is also true for humans, and it is the reason that there are no giants.  A human that is supported by his/her femur bones. These bones are the upper thigh bones and they are the widest and largest bones in the body.  Just like a stone pillar, the weight they can bear before they snap, is proportional to their cross-section.  When you increase the dimensions by twofold, the cross-section of the femurs increase by a factor of four, while the weight of the human increases eightfold.  Sooner or later, if you keep making a proportionally larger human, he will collapse under his own weight.

A great article - though a bit advanced - explains these ideas with more examples and lots of references to Galileo who first wrote about these ideas with scientific applications in mind. 

Problems:

1. In the metric system, a liter of water fills exactly a volume of 10 x 10 x 10 centimeters, and weighs exactly one kilogram.  How much does the water in a 10 x 10 x 10 meter swimming pool weigh?  Give your answer in kilograms, and then convert to pounds using 1 kg = 2.2 pounds.
2. If you drop a ball from a cliff, it will fall 16t² feet after t seconds.  How many feet does it fall after 4 seconds?  If you throw the ball upwards at 60 feet/second (40 mph), and you call the distance above the cliff positive, and below the cliff negative, the foot marker the ball is at after t seconds is:  60t - 16t².  Estimate how many seconds it will take before the ball comes back down, and passes you at the lip of the cliff.  The ball eventually succombs to gravity because gravity pulls at a quadratic rate while your arm throws at a linear rate.
   
3. Assume that the cross-section of the femurs of a 200 pound man can together support 400 pounds before they snap (ouch!).  By what factor can we expand this man before his femurs would be too weak to support his weight? 
   
4. An ant gets oxygen through its outer surface (no lungs), and its need for oxygen is directly proportional to its weight.  Let's say an ant needs 10 ml per second of oxygen to survive, and it gets 50 ml per second through its "skin" or outer surface.  If you make the ant twice as large in every dimension, then how much oxygen per second will it need to survive?  How much oxygen will it get per second?  Same question for 3x as large.  How much bigger can you make the ant before it suffocates?  This is why there is no such thing as giant ants.  By the way, lungs are designed the way they are to give an amazing surface area for the transfer of oxygen.  An adult human lung has a surface area of 100 square meters.  In contrast the surface area of an adult human's skin is only about 2 square meters.  This is why we breathe through our lungs and not our skin!
   

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