volume and surface area-universal...
TRANSCRIPT
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-&,-%&.%'A*MOTIVATING CALCULUS WITH BIOLOGY 3
Figure 1. A Cubical Critter
horse splashes. For the resistance presented to movement by the air is proportional to thesurface of the moving object. Divide an animal’s length, breadth, and height each by ten;its weight is reduced to a thousandth, but its surface only to a hundredth. So the resistanceto falling in the case of the small animal is relatively ten times greater than the drivingforce.
An insect, therefore, is not afraid of gravity; it can fall without danger, and can cling tothe ceiling with remarkably little trouble. It can go in for elegant and fantastic forms ofsupport like that of the daddy-longlegs. But there is a force which is as formidable to aninsect as gravitation to a mammal. This is surface tension. A man coming out of a bathcarries with him a film of water of about one-fiftieth of an inch in thickness. This weighsroughly a pound. A wet mouse has to carry about its own weight of water. A wet fly has tolift many times its own weight and, as everyone knows, a fly once wetted by water or anyother liquid is in a very serious position indeed. An insect going for a drink is in as greatdanger as a man leaning out over a precipice in search of food. If it once falls into the gripof the surface tension of the water that is to say, gets wet it is likely to remain so until itdrowns. A few insects, such as water-beetles, contrive to be unwettable; the majority keepwell away from their drink by means of a long proboscis. ”
To flush out this comments mathematically, we can begin simply by viewing all organisms as “cubicalcritters” that are characterized solely be their size differences (Fig. 1). If L is the length of one side of thecubical critter, then it has a surface area of 6L2 cm2,and a volume of L3m3. Furthermore, if we assumethat these cubical critters are “ugly bags of mostly water,” 2 then a critter of length L weighs m = L3
grams. Hence, surface area is proportional to m2/3 and the ratio of mass to surface area is proportional tom1/3. Hence, the larger you are, the harder you fall. Conversely, the ratio of surface area to mass is m−1/3.Therefore, the smaller you are, the more water weight you carry per unit biomass when you get wet.
There are many data sets which can be modeled by scaling laws. For instance, one of my favorite data setsis the mass lifted by an olympic weightlifter versus his body mass (Fig 2). Assuming Olympic weightlifters
2Star Trek fans may remember this line as an alien’s description of humans that are mostly water encased in a bag of skin.The ”ugly” part is a matter of extraterrestrial taste.
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MOTIVATING CALCULUS WITH BIOLOGY 3
Figure 1. A Cubical Critter
horse splashes. For the resistance presented to movement by the air is proportional to thesurface of the moving object. Divide an animal’s length, breadth, and height each by ten;its weight is reduced to a thousandth, but its surface only to a hundredth. So the resistanceto falling in the case of the small animal is relatively ten times greater than the drivingforce.
An insect, therefore, is not afraid of gravity; it can fall without danger, and can cling tothe ceiling with remarkably little trouble. It can go in for elegant and fantastic forms ofsupport like that of the daddy-longlegs. But there is a force which is as formidable to aninsect as gravitation to a mammal. This is surface tension. A man coming out of a bathcarries with him a film of water of about one-fiftieth of an inch in thickness. This weighsroughly a pound. A wet mouse has to carry about its own weight of water. A wet fly has tolift many times its own weight and, as everyone knows, a fly once wetted by water or anyother liquid is in a very serious position indeed. An insect going for a drink is in as greatdanger as a man leaning out over a precipice in search of food. If it once falls into the gripof the surface tension of the water that is to say, gets wet it is likely to remain so until itdrowns. A few insects, such as water-beetles, contrive to be unwettable; the majority keepwell away from their drink by means of a long proboscis. ”
To flush out this comments mathematically, we can begin simply by viewing all organisms as “cubicalcritters” that are characterized solely be their size differences (Fig. 1). If L is the length of one side of thecubical critter, then it has a surface area of 6L2 cm2,and a volume of L3m3. Furthermore, if we assumethat these cubical critters are “ugly bags of mostly water,” 2 then a critter of length L weighs m = L3
grams. Hence, surface area is proportional to m2/3 and the ratio of mass to surface area is proportional tom1/3. Hence, the larger you are, the harder you fall. Conversely, the ratio of surface area to mass is m−1/3.Therefore, the smaller you are, the more water weight you carry per unit biomass when you get wet.
There are many data sets which can be modeled by scaling laws. For instance, one of my favorite data setsis the mass lifted by an olympic weightlifter versus his body mass (Fig 2). Assuming Olympic weightlifters
2Star Trek fans may remember this line as an alien’s description of humans that are mostly water encased in a bag of skin.The ”ugly” part is a matter of extraterrestrial taste.
MOTIVATING CALCULUS WITH BIOLOGY 3
Figure 1. A Cubical Critter
horse splashes. For the resistance presented to movement by the air is proportional to thesurface of the moving object. Divide an animal’s length, breadth, and height each by ten;its weight is reduced to a thousandth, but its surface only to a hundredth. So the resistanceto falling in the case of the small animal is relatively ten times greater than the drivingforce.
An insect, therefore, is not afraid of gravity; it can fall without danger, and can cling tothe ceiling with remarkably little trouble. It can go in for elegant and fantastic forms ofsupport like that of the daddy-longlegs. But there is a force which is as formidable to aninsect as gravitation to a mammal. This is surface tension. A man coming out of a bathcarries with him a film of water of about one-fiftieth of an inch in thickness. This weighsroughly a pound. A wet mouse has to carry about its own weight of water. A wet fly has tolift many times its own weight and, as everyone knows, a fly once wetted by water or anyother liquid is in a very serious position indeed. An insect going for a drink is in as greatdanger as a man leaning out over a precipice in search of food. If it once falls into the gripof the surface tension of the water that is to say, gets wet it is likely to remain so until itdrowns. A few insects, such as water-beetles, contrive to be unwettable; the majority keepwell away from their drink by means of a long proboscis. ”
To flush out this comments mathematically, we can begin simply by viewing all organisms as “cubicalcritters” that are characterized solely be their size differences (Fig. 1). If L is the length of one side of thecubical critter, then it has a surface area of 6L2 cm2,and a volume of L3m3. Furthermore, if we assumethat these cubical critters are “ugly bags of mostly water,” 2 then a critter of length L weighs m = L3
grams. Hence, surface area is proportional to m2/3 and the ratio of mass to surface area is proportional tom1/3. Hence, the larger you are, the harder you fall. Conversely, the ratio of surface area to mass is m−1/3.Therefore, the smaller you are, the more water weight you carry per unit biomass when you get wet.
There are many data sets which can be modeled by scaling laws. For instance, one of my favorite data setsis the mass lifted by an olympic weightlifter versus his body mass (Fig 2). Assuming Olympic weightlifters
2Star Trek fans may remember this line as an alien’s description of humans that are mostly water encased in a bag of skin.The ”ugly” part is a matter of extraterrestrial taste.
MOTIVATING CALCULUS WITH BIOLOGY 3
Figure 1. A Cubical Critter
horse splashes. For the resistance presented to movement by the air is proportional to thesurface of the moving object. Divide an animal’s length, breadth, and height each by ten;its weight is reduced to a thousandth, but its surface only to a hundredth. So the resistanceto falling in the case of the small animal is relatively ten times greater than the drivingforce.
An insect, therefore, is not afraid of gravity; it can fall without danger, and can cling tothe ceiling with remarkably little trouble. It can go in for elegant and fantastic forms ofsupport like that of the daddy-longlegs. But there is a force which is as formidable to aninsect as gravitation to a mammal. This is surface tension. A man coming out of a bathcarries with him a film of water of about one-fiftieth of an inch in thickness. This weighsroughly a pound. A wet mouse has to carry about its own weight of water. A wet fly has tolift many times its own weight and, as everyone knows, a fly once wetted by water or anyother liquid is in a very serious position indeed. An insect going for a drink is in as greatdanger as a man leaning out over a precipice in search of food. If it once falls into the gripof the surface tension of the water that is to say, gets wet it is likely to remain so until itdrowns. A few insects, such as water-beetles, contrive to be unwettable; the majority keepwell away from their drink by means of a long proboscis. ”
To flush out this comments mathematically, we can begin simply by viewing all organisms as “cubicalcritters” that are characterized solely be their size differences (Fig. 1). If L is the length of one side of thecubical critter, then it has a surface area of 6L2 cm2,and a volume of L3m3. Furthermore, if we assumethat these cubical critters are “ugly bags of mostly water,” 2 then a critter of length L weighs m = L3
grams. Hence, surface area is proportional to m2/3 and the ratio of mass to surface area is proportional tom1/3. Hence, the larger you are, the harder you fall. Conversely, the ratio of surface area to mass is m−1/3.Therefore, the smaller you are, the more water weight you carry per unit biomass when you get wet.
There are many data sets which can be modeled by scaling laws. For instance, one of my favorite data setsis the mass lifted by an olympic weightlifter versus his body mass (Fig 2). Assuming Olympic weightlifters
2Star Trek fans may remember this line as an alien’s description of humans that are mostly water encased in a bag of skin.The ”ugly” part is a matter of extraterrestrial taste.
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MOTIVATING CALCULUS WITH BIOLOGY 3
Figure 1. A Cubical Critter
horse splashes. For the resistance presented to movement by the air is proportional to thesurface of the moving object. Divide an animal’s length, breadth, and height each by ten;its weight is reduced to a thousandth, but its surface only to a hundredth. So the resistanceto falling in the case of the small animal is relatively ten times greater than the drivingforce.
An insect, therefore, is not afraid of gravity; it can fall without danger, and can cling tothe ceiling with remarkably little trouble. It can go in for elegant and fantastic forms ofsupport like that of the daddy-longlegs. But there is a force which is as formidable to aninsect as gravitation to a mammal. This is surface tension. A man coming out of a bathcarries with him a film of water of about one-fiftieth of an inch in thickness. This weighsroughly a pound. A wet mouse has to carry about its own weight of water. A wet fly has tolift many times its own weight and, as everyone knows, a fly once wetted by water or anyother liquid is in a very serious position indeed. An insect going for a drink is in as greatdanger as a man leaning out over a precipice in search of food. If it once falls into the gripof the surface tension of the water that is to say, gets wet it is likely to remain so until itdrowns. A few insects, such as water-beetles, contrive to be unwettable; the majority keepwell away from their drink by means of a long proboscis. ”
To flush out this comments mathematically, we can begin simply by viewing all organisms as “cubicalcritters” that are characterized solely be their size differences (Fig. 1). If L is the length of one side of thecubical critter, then it has a surface area of 6L2 cm2,and a volume of L3m3. Furthermore, if we assumethat these cubical critters are “ugly bags of mostly water,” 2 then a critter of length L weighs m = L3
grams. Hence, surface area is proportional to m2/3 and the ratio of mass to surface area is proportional tom1/3. Hence, the larger you are, the harder you fall. Conversely, the ratio of surface area to mass is m−1/3.Therefore, the smaller you are, the more water weight you carry per unit biomass when you get wet.
There are many data sets which can be modeled by scaling laws. For instance, one of my favorite data setsis the mass lifted by an olympic weightlifter versus his body mass (Fig 2). Assuming Olympic weightlifters
2Star Trek fans may remember this line as an alien’s description of humans that are mostly water encased in a bag of skin.The ”ugly” part is a matter of extraterrestrial taste.
MOTIVATING CALCULUS WITH BIOLOGY 3
Figure 1. A Cubical Critter
horse splashes. For the resistance presented to movement by the air is proportional to thesurface of the moving object. Divide an animal’s length, breadth, and height each by ten;its weight is reduced to a thousandth, but its surface only to a hundredth. So the resistanceto falling in the case of the small animal is relatively ten times greater than the drivingforce.
An insect, therefore, is not afraid of gravity; it can fall without danger, and can cling tothe ceiling with remarkably little trouble. It can go in for elegant and fantastic forms ofsupport like that of the daddy-longlegs. But there is a force which is as formidable to aninsect as gravitation to a mammal. This is surface tension. A man coming out of a bathcarries with him a film of water of about one-fiftieth of an inch in thickness. This weighsroughly a pound. A wet mouse has to carry about its own weight of water. A wet fly has tolift many times its own weight and, as everyone knows, a fly once wetted by water or anyother liquid is in a very serious position indeed. An insect going for a drink is in as greatdanger as a man leaning out over a precipice in search of food. If it once falls into the gripof the surface tension of the water that is to say, gets wet it is likely to remain so until itdrowns. A few insects, such as water-beetles, contrive to be unwettable; the majority keepwell away from their drink by means of a long proboscis. ”
To flush out this comments mathematically, we can begin simply by viewing all organisms as “cubicalcritters” that are characterized solely be their size differences (Fig. 1). If L is the length of one side of thecubical critter, then it has a surface area of 6L2 cm2,and a volume of L3m3. Furthermore, if we assumethat these cubical critters are “ugly bags of mostly water,” 2 then a critter of length L weighs m = L3
grams. Hence, surface area is proportional to m2/3 and the ratio of mass to surface area is proportional tom1/3. Hence, the larger you are, the harder you fall. Conversely, the ratio of surface area to mass is m−1/3.Therefore, the smaller you are, the more water weight you carry per unit biomass when you get wet.
There are many data sets which can be modeled by scaling laws. For instance, one of my favorite data setsis the mass lifted by an olympic weightlifter versus his body mass (Fig 2). Assuming Olympic weightlifters
2Star Trek fans may remember this line as an alien’s description of humans that are mostly water encased in a bag of skin.The ”ugly” part is a matter of extraterrestrial taste.
MOTIVATING CALCULUS WITH BIOLOGY 3
Figure 1. A Cubical Critter
horse splashes. For the resistance presented to movement by the air is proportional to thesurface of the moving object. Divide an animal’s length, breadth, and height each by ten;its weight is reduced to a thousandth, but its surface only to a hundredth. So the resistanceto falling in the case of the small animal is relatively ten times greater than the drivingforce.
An insect, therefore, is not afraid of gravity; it can fall without danger, and can cling tothe ceiling with remarkably little trouble. It can go in for elegant and fantastic forms ofsupport like that of the daddy-longlegs. But there is a force which is as formidable to aninsect as gravitation to a mammal. This is surface tension. A man coming out of a bathcarries with him a film of water of about one-fiftieth of an inch in thickness. This weighsroughly a pound. A wet mouse has to carry about its own weight of water. A wet fly has tolift many times its own weight and, as everyone knows, a fly once wetted by water or anyother liquid is in a very serious position indeed. An insect going for a drink is in as greatdanger as a man leaning out over a precipice in search of food. If it once falls into the gripof the surface tension of the water that is to say, gets wet it is likely to remain so until itdrowns. A few insects, such as water-beetles, contrive to be unwettable; the majority keepwell away from their drink by means of a long proboscis. ”
To flush out this comments mathematically, we can begin simply by viewing all organisms as “cubicalcritters” that are characterized solely be their size differences (Fig. 1). If L is the length of one side of thecubical critter, then it has a surface area of 6L2 cm2,and a volume of L3m3. Furthermore, if we assumethat these cubical critters are “ugly bags of mostly water,” 2 then a critter of length L weighs m = L3
grams. Hence, surface area is proportional to m2/3 and the ratio of mass to surface area is proportional tom1/3. Hence, the larger you are, the harder you fall. Conversely, the ratio of surface area to mass is m−1/3.Therefore, the smaller you are, the more water weight you carry per unit biomass when you get wet.
There are many data sets which can be modeled by scaling laws. For instance, one of my favorite data setsis the mass lifted by an olympic weightlifter versus his body mass (Fig 2). Assuming Olympic weightlifters
2Star Trek fans may remember this line as an alien’s description of humans that are mostly water encased in a bag of skin.The ”ugly” part is a matter of extraterrestrial taste.
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MOTIVATING CALCULUS WITH BIOLOGY 3
Figure 1. A Cubical Critter
horse splashes. For the resistance presented to movement by the air is proportional to thesurface of the moving object. Divide an animal’s length, breadth, and height each by ten;its weight is reduced to a thousandth, but its surface only to a hundredth. So the resistanceto falling in the case of the small animal is relatively ten times greater than the drivingforce.
An insect, therefore, is not afraid of gravity; it can fall without danger, and can cling tothe ceiling with remarkably little trouble. It can go in for elegant and fantastic forms ofsupport like that of the daddy-longlegs. But there is a force which is as formidable to aninsect as gravitation to a mammal. This is surface tension. A man coming out of a bathcarries with him a film of water of about one-fiftieth of an inch in thickness. This weighsroughly a pound. A wet mouse has to carry about its own weight of water. A wet fly has tolift many times its own weight and, as everyone knows, a fly once wetted by water or anyother liquid is in a very serious position indeed. An insect going for a drink is in as greatdanger as a man leaning out over a precipice in search of food. If it once falls into the gripof the surface tension of the water that is to say, gets wet it is likely to remain so until itdrowns. A few insects, such as water-beetles, contrive to be unwettable; the majority keepwell away from their drink by means of a long proboscis. ”
To flush out this comments mathematically, we can begin simply by viewing all organisms as “cubicalcritters” that are characterized solely be their size differences (Fig. 1). If L is the length of one side of thecubical critter, then it has a surface area of 6L2 cm2,and a volume of L3m3. Furthermore, if we assumethat these cubical critters are “ugly bags of mostly water,” 2 then a critter of length L weighs m = L3
grams. Hence, surface area is proportional to m2/3 and the ratio of mass to surface area is proportional tom1/3. Hence, the larger you are, the harder you fall. Conversely, the ratio of surface area to mass is m−1/3.Therefore, the smaller you are, the more water weight you carry per unit biomass when you get wet.
There are many data sets which can be modeled by scaling laws. For instance, one of my favorite data setsis the mass lifted by an olympic weightlifter versus his body mass (Fig 2). Assuming Olympic weightlifters
2Star Trek fans may remember this line as an alien’s description of humans that are mostly water encased in a bag of skin.The ”ugly” part is a matter of extraterrestrial taste.
MOTIVATING CALCULUS WITH BIOLOGY 3
Figure 1. A Cubical Critter
horse splashes. For the resistance presented to movement by the air is proportional to thesurface of the moving object. Divide an animal’s length, breadth, and height each by ten;its weight is reduced to a thousandth, but its surface only to a hundredth. So the resistanceto falling in the case of the small animal is relatively ten times greater than the drivingforce.
An insect, therefore, is not afraid of gravity; it can fall without danger, and can cling tothe ceiling with remarkably little trouble. It can go in for elegant and fantastic forms ofsupport like that of the daddy-longlegs. But there is a force which is as formidable to aninsect as gravitation to a mammal. This is surface tension. A man coming out of a bathcarries with him a film of water of about one-fiftieth of an inch in thickness. This weighsroughly a pound. A wet mouse has to carry about its own weight of water. A wet fly has tolift many times its own weight and, as everyone knows, a fly once wetted by water or anyother liquid is in a very serious position indeed. An insect going for a drink is in as greatdanger as a man leaning out over a precipice in search of food. If it once falls into the gripof the surface tension of the water that is to say, gets wet it is likely to remain so until itdrowns. A few insects, such as water-beetles, contrive to be unwettable; the majority keepwell away from their drink by means of a long proboscis. ”
To flush out this comments mathematically, we can begin simply by viewing all organisms as “cubicalcritters” that are characterized solely be their size differences (Fig. 1). If L is the length of one side of thecubical critter, then it has a surface area of 6L2 cm2,and a volume of L3m3. Furthermore, if we assumethat these cubical critters are “ugly bags of mostly water,” 2 then a critter of length L weighs m = L3
grams. Hence, surface area is proportional to m2/3 and the ratio of mass to surface area is proportional tom1/3. Hence, the larger you are, the harder you fall. Conversely, the ratio of surface area to mass is m−1/3.Therefore, the smaller you are, the more water weight you carry per unit biomass when you get wet.
There are many data sets which can be modeled by scaling laws. For instance, one of my favorite data setsis the mass lifted by an olympic weightlifter versus his body mass (Fig 2). Assuming Olympic weightlifters
2Star Trek fans may remember this line as an alien’s description of humans that are mostly water encased in a bag of skin.The ”ugly” part is a matter of extraterrestrial taste.
MOTIVATING CALCULUS WITH BIOLOGY 3
Figure 1. A Cubical Critter
horse splashes. For the resistance presented to movement by the air is proportional to thesurface of the moving object. Divide an animal’s length, breadth, and height each by ten;its weight is reduced to a thousandth, but its surface only to a hundredth. So the resistanceto falling in the case of the small animal is relatively ten times greater than the drivingforce.
An insect, therefore, is not afraid of gravity; it can fall without danger, and can cling tothe ceiling with remarkably little trouble. It can go in for elegant and fantastic forms ofsupport like that of the daddy-longlegs. But there is a force which is as formidable to aninsect as gravitation to a mammal. This is surface tension. A man coming out of a bathcarries with him a film of water of about one-fiftieth of an inch in thickness. This weighsroughly a pound. A wet mouse has to carry about its own weight of water. A wet fly has tolift many times its own weight and, as everyone knows, a fly once wetted by water or anyother liquid is in a very serious position indeed. An insect going for a drink is in as greatdanger as a man leaning out over a precipice in search of food. If it once falls into the gripof the surface tension of the water that is to say, gets wet it is likely to remain so until itdrowns. A few insects, such as water-beetles, contrive to be unwettable; the majority keepwell away from their drink by means of a long proboscis. ”
To flush out this comments mathematically, we can begin simply by viewing all organisms as “cubicalcritters” that are characterized solely be their size differences (Fig. 1). If L is the length of one side of thecubical critter, then it has a surface area of 6L2 cm2,and a volume of L3m3. Furthermore, if we assumethat these cubical critters are “ugly bags of mostly water,” 2 then a critter of length L weighs m = L3
grams. Hence, surface area is proportional to m2/3 and the ratio of mass to surface area is proportional tom1/3. Hence, the larger you are, the harder you fall. Conversely, the ratio of surface area to mass is m−1/3.Therefore, the smaller you are, the more water weight you carry per unit biomass when you get wet.
There are many data sets which can be modeled by scaling laws. For instance, one of my favorite data setsis the mass lifted by an olympic weightlifter versus his body mass (Fig 2). Assuming Olympic weightlifters
2Star Trek fans may remember this line as an alien’s description of humans that are mostly water encased in a bag of skin.The ”ugly” part is a matter of extraterrestrial taste.
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-&,-%&.%'A*MOTIVATING CALCULUS WITH BIOLOGY 3
Figure 1. A Cubical Critter
horse splashes. For the resistance presented to movement by the air is proportional to thesurface of the moving object. Divide an animal’s length, breadth, and height each by ten;its weight is reduced to a thousandth, but its surface only to a hundredth. So the resistanceto falling in the case of the small animal is relatively ten times greater than the drivingforce.
An insect, therefore, is not afraid of gravity; it can fall without danger, and can cling tothe ceiling with remarkably little trouble. It can go in for elegant and fantastic forms ofsupport like that of the daddy-longlegs. But there is a force which is as formidable to aninsect as gravitation to a mammal. This is surface tension. A man coming out of a bathcarries with him a film of water of about one-fiftieth of an inch in thickness. This weighsroughly a pound. A wet mouse has to carry about its own weight of water. A wet fly has tolift many times its own weight and, as everyone knows, a fly once wetted by water or anyother liquid is in a very serious position indeed. An insect going for a drink is in as greatdanger as a man leaning out over a precipice in search of food. If it once falls into the gripof the surface tension of the water that is to say, gets wet it is likely to remain so until itdrowns. A few insects, such as water-beetles, contrive to be unwettable; the majority keepwell away from their drink by means of a long proboscis. ”
To flush out this comments mathematically, we can begin simply by viewing all organisms as “cubicalcritters” that are characterized solely be their size differences (Fig. 1). If L is the length of one side of thecubical critter, then it has a surface area of 6L2 cm2,and a volume of L3m3. Furthermore, if we assumethat these cubical critters are “ugly bags of mostly water,” 2 then a critter of length L weighs m = L3
grams. Hence, surface area is proportional to m2/3 and the ratio of mass to surface area is proportional tom1/3. Hence, the larger you are, the harder you fall. Conversely, the ratio of surface area to mass is m−1/3.Therefore, the smaller you are, the more water weight you carry per unit biomass when you get wet.
There are many data sets which can be modeled by scaling laws. For instance, one of my favorite data setsis the mass lifted by an olympic weightlifter versus his body mass (Fig 2). Assuming Olympic weightlifters
2Star Trek fans may remember this line as an alien’s description of humans that are mostly water encased in a bag of skin.The ”ugly” part is a matter of extraterrestrial taste.
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