Chapter 18: Growth and FormLesson Plan

Size and Growth

Geometric Similarity

How Much Is That In . . . ?

Scaling a Mountain

Sorry, No King Kongs

Dimension Tension

How to Grow

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Mathematical Literacy in Today’s World, 7th

ed.

For All Practical Purposes

© 2006, W.H. Freeman and Company

Chapter 18: Growth and Form Size and Growth

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Size and Growth Understanding the

underlying principles of scaling will help to appreciate why objects and living creatures in the world have the particular shapes and sizes that fit their function and survival.

Problem of Scale How every living species survives by adapting at the different

sizes from the beginning of life to the final size of a mature adult.

Chapter 18: Growth and Form Geometric Similarity

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Geometric Similarity The mathematical idea that two

objects are geometrically similar if the corresponding linear dimensions have the same factor of proportionality.

Example: To enlarge a photo, it is enlarged by the same factor in

Geometrically Similar – When two objects have the same shape, regardless of the materials of which they are made. both the horizontal and vertical directions—in fact, in any direction (such

as diagonal).

Linear Scaling Factor The number by which each linear dimension of an object is multiplied

when it is scaled up or down. The linear scaling factor of two geometrically similar objects is the ratio

of the length of any part of the second to the corresponding part of the first. See figure 18.2 on p. 665.

Chapter 18: Growth and Form Geometric Similarity

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How Area Scales The area of a scaled-up object goes up with the square of the linear

scaling factor. We symbolize the relationship between the area A and the linear

scaling factor L by:

A L2

where the symbol is read as “is proportional to” or “scales as.” How Volume Scales

The volume of a scaled-up object goes up with the cube of the linear scaling factor.

Denoting the volume by V, we can write:

V L3

Example: the length of each side of cube A is 2 inches and the length of each side of cube B is 12 inches.

What is the scaling factor of the enlargement from cube A to cube B?

What is the ratio of the diagonal of cube B to the diagonal of cube A?

What is the ratio of the surface area of the large cube B to that of the small cube A?

If the small cube A weighs 3 ounces, how much does the large cube B weigh?

Chapter 18: Growth and Form Geometric Similarity

Chapter 18: Growth and Form Geometric Similarity

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The Language of Growth, Enlargement, DecreaseHere are the correct ways to discuss meanings of percentages:

“x% of A” or “x% as large as” means × A.

“x% more than A” means A plus x% of A, in other words, (1 + ) × A.Saying that A has “increased by x%” means the same thing.

“x% less than A” means A minus x% of A, in other words, (1 − ) × A.Saying that A has “decreased by x%” means the same thing.

Note the following meanings: The terms of, times, and as much as refer to multiplication of the original amount. The terms more, larger, and greater refer to adding to the original amount. Do not use times with more or with less.

X 100

X 100

X 100

Chapter 18: Growth and Form How Much Is That In…?

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U.S. Customary System – United State’s measurement system

Distance1 mile (mi) = 1760 yards (yd) = 5280 feet (ft) = 63,360 inches (in)1 yard (yd) = 3 feet (ft) = 36 inches (in)1 foot (ft) = 12 inches (in)

Area1 square mile = 1 mi × 1 mi = 5280 ft × 5280 ft1 square mile = 640 acres1 acre = 43,560 ft2

Volume1 cubic mile = 1 mi × 1 mi × 1 mi = (5280 ft )3

1 U.S. gallon (gal) = 4 U.S. quarts (qt) = 231 in3, exactlyMass

1 ton (t) = 2000 pounds (lb)

Chapter 18: Growth and Form How Much Is That In…?

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Metric System – The rest of the world uses the metric system in science, industry, and commerce (meter for length; kilogram for mass).

Distance1 meter (m) = 100 centimeters (cm)1 kilometer (km) = 1000 meters (m) = 100,000 cm = 1 × 105 cm

Area1 square meter (m2) = 1 m × 1 m

= 100 cm × 100cm = 10,000 cm2 = 1 × 104 cm2

1 hectare (ha) = 10,000m2

Volume1 liter (L) = 1000 cm3 = 0.001 m3

1 cubic meter (m3) = 1 m × 1 m × 1 m = (100cm)3 = 1,000,000 cm3 = 1 × 106 cm3 (or

cc)Mass

1 kilogram (kg) = 1000 grams (g)

Chapter 18: Growth and Form How Much Is That In…?

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Conversions Between U.S. Customary and Metric SystemsDistance

1 in = 2.54 cm, exactly1 ft = 12 in = 12 × 2.54 cm = 30.48 cm = 0.3048 m, exactly1 yd = 0.9144m, exactly1 mi = 5280 ft = 5280 × 30.48 cm = 160,934.4 cm ≈ 1.61 km

Area1 hectare (ha) ≈ 2.47 acres

Volume1 cubic meter (m3) = 1000 liters ≈ 264.2 U.S. gallons ≈ 35.31 ft3

1 liter (L) = 1000 cm3 ≈ 1.057 U.S. quarts (qt) ≈ 0.2642 U.S. gallonsMass

1 lb = 0.45359237 kg, exactly1 kg ≈ 2.205 lb

Example: How much is 160 cm tall in feet? 160 cm × × ≈ 5.25 ft Units cancel out, leaving feet in the

answer.

1 in . 2.54 cm

1 ft 12 in

Examples:

A weight of 5 lbs is approximately the same as _____ kg.

A distance of 4 yards is the same as _____ feet.

A volume of 8 quarts is the same as _____ gallons.

An area of 2000 in2 is approximately the same as _____ m2.

Chapter 18: Growth and Form How Much Is That In…?

Chapter 18: Growth and Form Scaling a Mountain

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Scaling Up Objects Matter is determined by

its mass, which is its volume multiplied by its density.

Weight – Force under gravity.Pressure – Force per unit area,

P = W /A.Density – Mass per unit volume

density = mass/V. The force of gravity is what gives an object its weight. For all matter there is an upper limit to the amount of pressure or

weight that it can support without distortion. This limit is known as the crushing strength that is inherent in all materials.

As a result, there is a limit to which objects can be scaled up because the pressure increases proportionally to the scaling factor.

If the size increases enough, scaled up beyond what the material can hold, the object will buckle under its own weight.

To solve the problem of scale, it would be necessary to change the material composition (to one that has higher crushing strength), or change the shape of the object, or both.

Chapter 18: Growth and Form Scaling a Mountain

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A Mile-High Building? In 1956 Frank Lloyd Wright (1867–

1959) proposed a mile-high tower for the Chicago lakefront.

There are many limitations to how high to make a building:

Bending of the building in wind. More room for elevators, fire

escapes, heat, and air conditioning for the taller the building.

There could be limitations on human physiology, such as the difference in air pressure from top to bottom.

Scaling Up Objects The height of a mountain or building is limited to gravity,

its composition (material strength), and its shape. Crushing strength of steel = 7.5 million lb/ft2, and crushing

strength of granite = 4 million lb/ft2.

Chapter 18: Growth and Form Sorry, No King Kongs

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Sorry, No King Kongs The resistance of bone to crushing is not nearly as great as that of

steel or granite. This explains why there cannot be any King Kongs (unless they were made of steel or granite!).

How Tall Can a Tree Be? Galileo suggested that no tree could grow taller

than 300 ft, but he did not know of the sequoias on the West Coast of the United States.

The tallest tree today is 369 ft, and the tallest that was reliably measured was 413 ft.

Limitations of tree height: Root system must anchor the tree properly. The wood at the bottom could crush if there is too much

weight above. Limit to how far the tree can lift water and minerals from

the roots to the leaves.

Chapter 18: Growth and Form Dimension Tension

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Dimension Tension Large change in scale forces a change in either materials

(change to a stronger material) or form (may need to redesign the object).

The problem resulting from scaling up is the tension between weight and the need to support it.

Area-Volume Tension – The result of the fact that, as an object is scaled up, the volume increases faster than the surface area and faster than areas of cross sections.

Example: The diagram on the left shows the weight being distributed better by having two rows of cubes down on the floor level and cutting another row of cubes in half and distributing them across the top.

Chapter 18: Growth and Form How to Grow

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How to Grow Within narrow limits (up to a factor of 2), most creatures can grow

according to geometric similarity, or grow proportionally. However, most living things grow by a factor greater than 2.

Growth laws are much more complicated than for proportional growth. Computerized morphing techniques take a person’s picture and

change it smoothly, with different scalings for different parts of the face.

National Center for Missing and Exploited Children (NCMEC) uses computer programs to show age progression.

Age 7 Age 17

Image stretching is applied to a photograph to progress the age to obtain a rough idea of what a child may look like 10 years later.

Chapter 18: Growth and Form How to Grow

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Human Growth Basically, a human adult is not simply a scaled-up baby. Allometric Growth – Growth of the length of one feature at a rate

proportional to a power of the length of another. Before 9 months, the arm length increases relatively faster than height.

Proportional Growth – After 9 months, a change in pattern occurs. Growth occurs more proportionally, also known as isometric growth.

Different parts of the body scale geometrically, each with a different scale factor.

Babies’ eyes grow to perhaps twice their original size, while the arms grow to about four times their original size.

The proportions of the human body change with age. Notice that the baby’s head is larger in proportion to the rest of its body.