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1.4 Approximation  (Page 2/3)

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Solution

(1) Calculate the volume of a stack of 100 bills. The dimensions of a single bill are approximately 3 in. by 6 in. A stack of 100 of these is about 0.5 in. thick. So the total volume of a stack of 100 bills is:

volume of stack = length × width × height, volume of stack = 6 in . × 3 in . × 0 . 5 in . , volume of stack = 9 in . 3 . alignl { stack { size 12{"volume of stack"="length" times "width" times "height,"} {} #size 12{"volume of stack"="6 in" "." times "3 in" "." times 0 "." "5 in" "." ,} {} # size 12{"volume of stack"="9 in" "." rSup { size 8{3} } "." } {}} } {}

(2) Calculate the number of stacks. Note that a trillion dollars is equal to $ 1 × 10 12 , size 12{$1 times "10" rSup { size 8{"12"} } } {} and a stack of one-hundred $ 100 size 12{$"100"} {} bills is equal to $ 10 , 000 , size 12{$"10","000",} {} or $ 1 × 10 4 size 12{$1 times "10" rSup { size 8{4} } } {} . The number of stacks you will have is:

$ 1 × 10 12 ( a trillion dollars ) / $1 × 10 4 per stack = 1 × 10 8 stacks. size 12{$1 times "10" rSup { size 8{"12"} } \( "a trillion dollars" \) " / $1" times "10" rSup { size 8{4} } " per stack"=1 times "10""" lSup { size 8{8} } " stacks"} {}

(3) Calculate the area of a football field in square inches. The area of a football field is 100 yd × 50 yd, which gives 5, 000 yd 2 . size 12{5,"000 yd" rSup { size 8{2} } } {} Because we are working in inches, we need to convert square yards to square inches:

Area = 5,000 yd 2 × 3 ft 1 yd × 3 ft 1 yd × 12 in . 1 ft × 12 in . 1 ft = 6, 480 , 000 in . 2 , Area ≈ 6 × 10 6 in . 2 . alignl { stack { size 12{"Area"="5,000 yd" rSup { size 8{2} } times { {3" ft"} over {"1yd"} } times { {3" ft"} over {"1yd"} } times { {"12"" in" "." } over {"1 foot"} } times { {"12"" in" "." } over {"1 foot"} } =6,"480","000"" in" "." rSup { size 8{2} } ,} {} #"Area" approx 6 times "10" rSup { size 8{6} } " in" "." rSup { size 8{2} } "." {} } } {}

This conversion gives us 6 × 10 6 in . 2 size 12{6 times "10" rSup { size 8{6} } `"in" "." rSup { size 8{2} } } {} for the area of the field. (Note that we are using only one significant figure in these calculations.)

(4) Calculate the total volume of the bills. The volume of all the $ 100 size 12{$"100"} {} -bill stacks is 9 in . 3 / stack × 10 8  stacks = 9 × 10 8 in . 3 . size 12{9" in" "." rSup { size 8{3} } /"stack" times "10" rSup { size 8{8} } "stacks"=9 times "10" rSup { size 8{8} } " in" "." rSup { size 8{3} } } {}

(5) Calculate the height. To determine the height of the bills, use the equation:

volume of bills = area of field × height of money: Height of money = volume of bills area of field , Height of money = 9 × 10 8 in . 3 6 × 10 6 in. 2 = 1.33 × 10 2 in., Height of money ≈ 1 × 10 2 in. = 100 in. alignl { stack { size 12{"volume of bills"="area of field" times "height of money:"} {} #size 12{"Height of money"= { {"volume of bills"} over {"area of field"} } ,} {} # size 12{"Height of money"= { {9 times "10" rSup { size 8{8} } " in" "." rSup { size 8{3} } } over {6 times "10" rSup { size 8{6} } " in" "." rSup { size 8{2} } } } =1 "." "33" times "10" rSup { size 8{2} } " in" "." ,} {} #"Height of money" approx 1 times "10" rSup { size 8{2} } " in" "." ="100 in" "." {} } } {}

The height of the money will be about 100 in. high. Converting this value to feet gives

100 in . × 1 ft 12 in . = 8 . 33 ft ≈ 8 ft. size 12{"100 in" "." times { {"1ft"} over {"12 in" "." } } =8 "." "33 ft" approx "8 ft"} {}

Discussion

The final approximate value is much higher than the early estimate of 3 in., but the other early estimate of 10 ft (120 in.) was roughly correct. How did the approximation measure up to your first guess? What can this exercise tell you in terms of rough “guesstimates” versus carefully calculated approximations?

Using mental math and your understanding of fundamental units, approximate the area of a regulation basketball court. Describe the process you used to arrive at your final approximation.

An average male is about two meters tall. It would take approximately 15 men laid out end to end to cover the length, and about 7 to cover the width. That gives an approximate area of 420  m 2 size 12{"420"" m" rSup { size 8{2} } } {} .

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Summary

Scientists often approximate the values of quantities to perform calculations and analyze systems.

Problems&Exercises

How many heartbeats are there in a lifetime?

Sample answer: 2 × 10 9 size 12{2 times "10" rSup { size 8{9} } } {} heartbeats

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A generation is about one-third of a lifetime. Approximately how many generations have passed since the year 0 AD?

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How many times longer than the mean life of an extremely unstable atomic nucleus is the lifetime of a human? (Hint: The lifetime of an unstable atomic nucleus is on the order of 10 − 22  s size 12{"10" rSup { size 8{ - "22"} } " s"} {} .)

Sample answer: 2 × 10 31 size 12{2 times "10" rSup { size 8{"31"} } } {} if an average human lifetime is taken to be about 70 years.

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Calculate the approximate number of atoms in a bacterium. Assume that the average mass of an atom in the bacterium is ten times the mass of a hydrogen atom. (Hint: The mass of a hydrogen atom is on the order of 10 − 27  kg size 12{"10" rSup { size 8{ - "27"} } " kg"} {} and the mass of a bacterium is on the order of 10 − 15  kg. size 12{"10" rSup { size 8{ - "15"} } "kg"} {} )

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A magnified image of the bacterium Salmonella attacking a human cell. The bacterium is rod shaped and about zero point seven to one point five micrometers in diameter and two to five micrometers in length.
This color-enhanced photo shows Salmonella typhimurium (red) attacking human cells. These bacteria are commonly known for causing foodborne illness. Can you estimate the number of atoms in each bacterium? (credit: Rocky Mountain Laboratories, NIAID, NIH)

Approximately how many atoms thick is a cell membrane, assuming all atoms there average about twice the size of a hydrogen atom?

Sample answer: 50 atoms

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(a) What fraction of Earth’s diameter is the greatest ocean depth? (b) The greatest mountain height?

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(a) Calculate the number of cells in a hummingbird assuming the mass of an average cell is ten times the mass of a bacterium. (b) Making the same assumption, how many cells are there in a human?

Sample answers:

(a) 10 12 size 12{"10" rSup { size 8{"12"} } } {} cells/hummingbird

(b) 10 16 size 12{"10" rSup { size 8{"16"} } } {} cells/human

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Assuming one nerve impulse must end before another can begin, what is the maximum firing rate of a nerve in impulses per second?

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Questions & Answers

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Astronomy (from Ancient Greek ἀστρονομία (astronomía) 'science that studies the laws of the stars') is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution.
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No
Richard
the big bang theory is a theory which states that all matter was compressed together in one place the matter got so unstable it exploded releasing All its contents in the form of hydrogen
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solar Univers
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no
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bigbang is the beginning of the universe
Sokak
but thats just a theory
Sokak
nothing will happen, don't worry brother.
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GANGAIN Reply
these are Rocky substances between mars and jupiter
GANGAIN
Comets are cosmic snowballs of frozen gases , rock and dust that orbit the sun. They are mostly found between the orbits of Venus and Mercury.
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Gatjuol Reply
Because when astroid hit the Earth then a piece of elliptical shape of the earth was separated which is now called moon.
Hemen
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lidiya Reply
Did you mean eye sight or sea level
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oh sorry it's sea level
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according to the theory of astronomers why the moon is always appear in an elliptical orbit?
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hi !!! I am new in astronomy.... I have so many questions in mind .... all of scientists of the word they just give opinion only. but they never think true or false ... i respect all of them... I believes whole universe depending on true ...থিউরি
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there are many theories regarding this it's on you believe any theory that you think is true ex. eternal inflation theory, oscillation model theory, multiple universe theory the big bang theory etc.
Aarya
I think after Big Bang!
Michele
from where on earth could u observe all the stars during the during the course of an year
Karuna Reply
I think it couldn't possible on earth
Nagalakshmi
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Michele
is that so. the question was in the end of this chapter
Karuna
in theory, you could see them all from the equator (though over the course of a year, not at pne time). stars are measured in "declination", which is how far N or S of the equator (90* to -90*). Polaris is the North star, and is ALMOST 90* (+89*). So it would just barely creep over the horizon.
Christopher
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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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