1. Home
  2. College physics
  3. Introduction: the nature of
  4. Approximation

1.4 Approximation  (Page 2/3)

<< Chapter < Page
  College physics     Page 2 / 3
Chapter >> Page >

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} } } {} .

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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"} {} )

Got questions? Get instant answers now!
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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

(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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

Questions & Answers

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
which drugs can we use for ulcers
Ummi Reply
omeprazole
Kamara
what
Renee
what is this
Renee
is a drug
Kamara
of anti-ulcer
Kamara
Omeprazole Cimetidine / Tagament For the complicated once ulcer - kit
Patrick
what is the function of lymphatic system
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
asegid
to transport fluids fats proteins and lymphocytes to the blood stream as lymph
Adama
what is anatomy
Oyindarmola Reply
Anatomy is the identification and description of the structures of living things
Kamara
what's the difference between anatomy and physiology
Oyerinde Reply
Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.
AI-Robot
what is enzymes all about?
Mohammed Reply
Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems
Kamara
yes
Prince
how does the stomach protect itself from the damaging effects of HCl
Wulku Reply
little girl okay how does the stomach protect itself from the damaging effect of HCL
Wulku
it is because of the enzyme that the stomach produce that help the stomach from the damaging effect of HCL
Kamara
function of digestive system
Ali Reply
function of digestive
Ali
the diagram of the lungs
Adaeze Reply
what is the normal body temperature
Diya Reply
37 degrees selcius
Xolo
37°c
Stephanie
please why 37 degree selcius normal temperature
Mark
36.5
Simon
37°c
Iyogho
the normal temperature is 37°c or 98.6 °Fahrenheit is important for maintaining the homeostasis in the body the body regular this temperature through the process called thermoregulation which involves brain skin muscle and other organ working together to maintain stable internal temperature
Stephanie
37A c
Wulku
what is anaemia
Diya Reply
anaemia is the decrease in RBC count hemoglobin count and PVC count
Eniola
what is the pH of the vagina
Diya Reply
how does Lysin attack pathogens
Diya
acid
Mary
I information on anatomy position and digestive system and there enzyme
Elisha Reply
anatomy of the female external genitalia
Muhammad Reply
Organ Systems Of The Human Body (Continued) Organ Systems Of The Human Body (Continued)
Theophilus Reply
what's lochia albra
Kizito
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>
Practice Key Terms 1

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now
Start Quiz
Start FlashCards Download PDF Learn




Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College physics' conversation and receive update notifications?

Ask