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3.1 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} } } {} .

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

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

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.

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"} {} )

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

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

(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

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

Questions & Answers

distinguish between anatomy and physiology
Amina Reply
Anatomy is the study of internal structure of an organism while physiology is the study of the function/relationship of the body organs working together as a system in an organism.
adeyeye
distinguish between anatomy and physiology
Erny Reply
regional anatomy is the study of the body regionally
Ismail Reply
what is the meaning of regional anatomy
Aminat Reply
epithelial tissue: it covers the Hollow organs and body cavities
Esomchi Reply
in short way what those epithelial tissue mean
Zainab Reply
in short way what those epithelial tissue mean
Chizoba
What is the function of the skeleton
Lilias Reply
movement
Ogar
Locomotion
Ojo
support
Aishat
and body shape/form
Aishat
what is homeostasis?
Samuel Reply
what's physiology
AminchiSunday Reply
what is physiology
AminchiSunday
physically is the study of the function of the body
Najaatu
that is what I want ask
YAU
u are wright
YAU
pls what are the main treatment of hiccups
YAU
physiology is the study of the function of the body
Najaatu
hiccups happen when something irritates the nerves that course your diaphragm to contract
Najaatu
how did hypothalamus manege to control all activities of the various hormones
malual
what is protein
Abdulsalam
how can I treat pain a patient feels after eating meals
Namuli Reply
how do I treat a three year old baby of skin infection?
Okocha Reply
It depends on the type of infection. Bacterial, fungal, parasitic or viral?
schler
if you can share the sign ad symptoms of the skin infection then u geh the treatment cox they're different sign ad symptoms of skin infection with different treatment
Sa
the sign and symptoms of maleria
Abdulsalam
prostaglandin and fever
Maha Reply
yes
rayyanu
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rayyanu
prostaglandin E2 is the final mediator.
Lemlem
prostaglandin E2 is the final mediator of fever.
Lemlem
yes
Agabi
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Jediel
tissue.
Akoi
explain
Chizoba
Hi
Anya
,good evening
Anya
Discuss the differences between taste and flavor, including how other sensory inputs contribute to our  perception of flavor.
John Reply
taste refers to your understanding of the flavor . while flavor one The other hand is refers to sort of just a blend things.
Faith
While taste primarily relies on our taste buds, flavor involves a complex interplay between taste and aroma
Kamara
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Ummi Reply
omeprazole
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Renee
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Kamara
Omeprazole Cimetidine / Tagament For the complicated once ulcer - kit
Patrick
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Nency Reply
Not really sure
Eli
to drain extracellular fluid all over the body.
asegid
The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include: 1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of
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Source:  OpenStax, Physics 105: adventures in physics. OpenStax CNX. Dec 02, 2015 Download for free at http://legacy.cnx.org/content/col11916/1.1
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