Estimations and Approximations
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Estimations and Approximations

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Questions and Answers

What is the primary purpose of back-of-an-envelope calculations?

  • To replace detailed calculations in engineering tasks.
  • To ensure accuracy in every measurement.
  • To quickly estimate an answer for initial assessments. (correct)
  • To provide precise answers to calculations.

    • When estimating the Earth's diameter, which assumption is made?

  • Johannesburg and London are directly west of each other.
  • The flying time from Johannesburg to London is approximately 10 hours. (correct)
  • London is a major city in the southern hemisphere.
  • The circumference applies only to the polar region.

    • What does '≈' represent in estimations?

  • Approximately equal to (correct)
  • Exactly equal to
  • More or less equal to
  • Is equivalent to

    • Which of the following represents a significant figure consideration in estimations?

    <p>Considering the precision of measurements.</p> Signup and view all the answers

    When estimating how fast the Sun moves through the sky, which factor is not directly considered?

    <p>The speed of light.</p> Signup and view all the answers

    What is implied when stating that London is 'about a quarter of the way around the globe' from Johannesburg?

    <p>The Earth's circumference can be divided into quarters for estimation.</p> Signup and view all the answers

    Which of the following best describes the relationship between the circumference and diameter of the Earth in estimation?

    <p>The diameter can be derived from the circumference using a constant factor.</p> Signup and view all the answers

    In the context of estimations, what does the term 'subjective component' imply?

    <p>Estimations can be influenced by personal experiences and intuition.</p> Signup and view all the answers

    What is the primary recommendation for improving population density estimates in South Africa?

    <p>Reducing the area of South Africa by the areas of Lesotho and Swaziland</p> Signup and view all the answers

    In estimating the number of words to reach 1MB of data, what assumption is made about each character?

    <p>Each character is equivalent to 1 byte</p> Signup and view all the answers

    How many lines are assumed to be on one exam page for the hand-written estimate?

    <p>30 lines</p> Signup and view all the answers

    What is the relationship between the accuracy of assumptions and the estimated answers according to the content?

    <p>Accurate assumptions ensure precise estimates</p> Signup and view all the answers

    What is the estimated average number of words per line of handwriting?

    <p>10 words</p> Signup and view all the answers

    What angular difference is recommended for improving estimates related to Earth’s diameter?

    <p>78°</p> Signup and view all the answers

    What is the actual distance mentioned for estimating Earth's diameter?

    <p>10,496 km</p> Signup and view all the answers

    Which latitude consideration is made for estimating the Sun's overhead position in South Africa?

    <p>Tropic of Cancer</p> Signup and view all the answers

    What is noted about Cape Town's location in relation to Johannesburg?

    <p>It is further south.</p> Signup and view all the answers

    What is the approximate diameter of the Earth calculated in the methodology?

    <p>12,700 km</p> Signup and view all the answers

    What conclusion is drawn despite the inaccuracies in estimates for Earth’s diameter?

    <p>The estimates correlate well within 10%.</p> Signup and view all the answers

    What is the approximate speed of the Sun through the sky in km/h?

    <p>1,700 km/h</p> Signup and view all the answers

    From which campaign is the estimated population of South Africa derived?

    <p>49M energy-saving campaign</p> Signup and view all the answers

    How much time later does the sun set in Cape Town compared to Johannesburg?

    <p>36 minutes</p> Signup and view all the answers

    What assumption is made about the population distribution in South Africa?

    <p>Only major cities are populated.</p> Signup and view all the answers

    What is the angular difference in degrees between Johannesburg and London?

    <p>90°</p> Signup and view all the answers

    What methodology is described for estimating population density in South Africa?

    <p>Estimating the number of people and area.</p> Signup and view all the answers

    How is the circumference of the Earth determined based on the distance from Johannesburg to London?

    <p>By multiplying the distance by 4</p> Signup and view all the answers

    What factor is used to calculate the Earth's rotational speed?

    <p>Time taken for one complete revolution</p> Signup and view all the answers

    What error is included in the actual flying time to London?

    <p>Overlooking the time spent on maneuvering</p> Signup and view all the answers

    What is the approximate distance that separates London and Johannesburg horizontally?

    <p>3,200 km</p> Signup and view all the answers

    What is the estimated total number of words in 1MB based on average letters per word?

    <p>233,017 words</p> Signup and view all the answers

    How many hand-written words fit onto one page of an exam pad?

    <p>300 words</p> Signup and view all the answers

    What is the estimated number of exam pages required for 700,000 words?

    <p>700 pages</p> Signup and view all the answers

    How many megabytes is the dictionary estimated to be?

    <p>4.8 MB</p> Signup and view all the answers

    Which factor significantly impacts the number of hand-written exam pages estimated?

    <p>The quality of hand-writing</p> Signup and view all the answers

    What is the average number of letters in an English word used for these calculations?

    <p>5 letters</p> Signup and view all the answers

    If punctuation and spaces are not considered, how does this affect the word count?

    <p>Decreases the word count</p> Signup and view all the answers

    What is a recommended method for improving estimates of words in the dictionary?

    <p>Use a sample of one page and check total pages</p> Signup and view all the answers

    Study Notes

    Estimations and Approximations

    • Objective: Highlight the effectiveness of quick calculations for initial estimates and problem checks in various situations.
    • Key symbols explained:
      • "=" means exactly equal
      • "≈" denotes approximately equal
      • "≡" indicates equivalence
      • "∼" implies about or more or less
      • "±" represents positive and negative variability.
    • Guesstimates rely on common sense, experience, and subjective judgment.

    Example 1 – Earth's Circumference and Diameter

    • Circumference Estimation:
      • Distance from Johannesburg to London is estimated as 10,000 km using the formula: distance = speed × time (1,000 km/h × 10 h).
      • Earth's circumference is calculated from this distance: 40,000 km (4 × 10,000 km).
    • Diameter Calculation:
      • Diameter = Circumference / π ≈ 12,700 km.
    • Speed of Sun Through Sky:
      • Estimated as 1,700 km/h, derived from Earth's 24-hour rotation delivering 40,000 km distance.
    • Sunset Time in Cape Town:
      • Angular difference estimated to be about 9°, leading to a time difference of approximately 36 minutes, with a resulting sunset time of 18:06.

    Actual vs Estimated Values for Example 1:

    • Actual Earth circumference: 40,075 km; diameter: 12,756.3 km.
    • Actual speed of the Sun: 1,669.8 km/h.
    • Actual sunset time: 17:53.

    Recommendations for Improved Estimates:

    • Use accurate angular difference (78°) instead of a quarter-circle assumption (90°).
    • Consider the true distance flown (10,496 km) and actual cruising speeds of aircraft (e.g., Boeing 747 at 933 km/h).
    • Account for Earth's elliptical shape.

    Example 2 – Population Density of South Africa

    • Population Estimate:
      • ~49 million people based on historical data from the 2010 World Cup; urban assumption as ~4 million in Johannesburg.
    • Area Estimation:
      • Length of South Africa approximated at ~2,000 km.
    • Population Density Calculation:
      • Inferences based on populated urban settings, neglecting rural populations.

    Recommendations for Improved Population Estimates:

    • Adjust for the areas of neighboring countries like Lesotho and Swaziland.
    • Increase the sampling squares to better represent South Africa’s geographic shape.

    Example 3 – Text to Megabytes (MB)

    • Words in 1 MB:
      • Estimated as ≈210,000 words based on parameters that 1 character = 1 byte and average word length is 5 letters.
    • Exam Pages Calculation:
      • Approximately 700 hand-written pages required to fill 1 MB (300 words/page).
    • Pocket Oxford Dictionary Size Estimation:
      • Estimated to contain 1,000,000 words, resulting in ≈4.8 MB when typed.

    Actual vs Estimated Values for Example 3:

    • Actual words per MB: 233,017 words.
    • Dictionary size estimated between 4 MB and 6 MB depending on specific version details.

    General Discussion and Conclusions

    • Quality of assumptions greatly impacts estimate accuracy: "rubbish in = rubbish out."
    • Approaches for estimation should evolve based on sampling and verification for accuracy.

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    Related Documents

    FEBE1004A – EAD 1B Lecture Packages – Estimations and Approximations PDF

    Description

    This quiz explores the use of quick calculations for making estimations and checking problems across various scenarios. It covers key symbols used in estimation and provides examples, such as calculating the Earth's circumference and diameter. Test your understanding of these concepts and improve your guesstimation skills!

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