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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

how did you get 1640
Noor Reply
If auger is pair are the roots of equation x2+5x-3=0
Peter Reply
Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.
MATTHEW Reply
420
Sharon
from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph
George
Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28
Salma
Devon is 32 32​​ years older than his son, Milan. The sum of both their ages is 54 54​. Using the variables d d​ and m m​ to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.
Aaron Reply
find product (-6m+6) ( 3m²+4m-3)
SIMRAN Reply
-42m²+60m-18
Salma
what is the solution
bill
how did you arrive at this answer?
bill
-24m+3+3mÁ^2
Susan
i really want to learn
Amira
I only got 42 the rest i don't know how to solve it. Please i need help from anyone to help me improve my solving mathematics please
Amira
Hw did u arrive to this answer.
Aphelele
hi
Bajemah
-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18
Salma
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Jovelyn Reply
x=3-2y
Salma
y=x+3/2
Salma
Hi
Enock
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Nandala
3x-12y=18
Kelvin
please why isn't that the 0is in ten thousand place
Grace Reply
please why is it that the 0is in the place of ten thousand
Grace
Send the example to me here and let me see
Stephen
A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg
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how far
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cool u
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I am eliacin, I need your help in maths
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how can I help
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hmm can we speak here?
Amoon
however, may I ask you some questions about Algarba?
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what the last part of the problem mean?
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cameron Reply
Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.
mahnoor Reply
I'm guessing, but it's somewhere around $4335.00 I think
Lewis
12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00
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Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?
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d=r×t the equation would be 8/r+24/r+4=3 worked out
Sheirtina
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Source:  OpenStax, College physics for ap® courses. OpenStax CNX. Nov 04, 2016 Download for free at https://legacy.cnx.org/content/col11844/1.14
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