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By looking at a rectangular box, a carpenter estimates that the length of the box is between 2 to 2.1 meters, inclusive, the breadth is between 1 to 1.1 meters, inclusive and the height is between 2 to 2.1 centimeters, inclusive. If the actual length, breadth and height of the box do indeed fall within the respective ranges estimated by the carpenter, which of the following is the closest to the maximum possible magnitude of the percentage error that the carpenter can make in calculating the volume of the rectangular box?
  • a)
    1%
  • b)
    3%
  • c)
    10%
  • d)
    18%
  • e)
    22%
Correct answer is option 'E'. Can you explain this answer?
Verified Answer
By looking at a rectangular box, a carpenter estimates that the length...
Given:
  • 2 meters ≤ Estimated Length of the rectangular box, LE ≤ 2.1 meters
  • 1 meter ≤ Estimated Breadth of the rectangular box, BE ≤ 1.1 meters
  • 2 centimeters ≤ Estimated Height of the rectangular box, HE ≤ 2.1 centimeters
  • The actual L, B and H of the box fall within the respective estimated ranges
To find: Approx. value of the maximum % error in estimating the volume
Approach:
  1. Error Percentage in Volume calculation = 
    • The magnitude of the error percentage will be maximum when the magnitude of the numerator is maximum and denominator is minimum.
    • So, the case of Maximum possible error arises when:
      1. the actual volume is in fact the least possible, given the ranges of LE, BE and HE AND
      2. the carpenter calculates the volume using the maximum estimated values of LE, BE and HE
         
  1. So we’ll calculate the least possible value of the actual volume and the greatest possible value of the calculated volume
    • Note that in Volume calculations, we should convert the units of length and breadth into centimeters (to make the units of length, breadth and height uniform)
Working Out:
  • Finding least possible value of the actual volume
    • The least possible actual volume occurs when:
      • L = 2 meters = 200 centimeters
      • H = 2 centimeters
      • B = 1 meter = 100 centimeters
    • So, least possible actual volume = 200∗100∗2=40000 cm3
 
  • Finding greatest possible value of the calculated volume
    • The greatest possible calculated volume occurs when:
      • Estimated length used for the calculation = 2.1 meters = 210 centimeters
      • Estimated breadth used for the calculation = 1.1 meter = 110 centimeters
      • Estimated height used for the calculation = 2.1 centimeters
So, greatest possible calculated volume = = 210*110*2.1 = =210∗11∗21 cm3
Finding the maximum % error
  • So, Maximum error Percentage in Volume calculation =
Looking at the answer choices, we see that the answer choice which is closest to the value we obtained is Option E (22%) So, Option E is the correct answer
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By looking at a rectangular box, a carpenter estimates that the length of the box is between 2 to 2.1 meters, inclusive, the breadth is between 1 to 1.1 meters, inclusive and the height is between 2 to 2.1 centimeters, inclusive. If the actual length, breadth and height of the box do indeed fall within the respective ranges estimated by the carpenter, which of the following is the closest to the maximum possible magnitude of the percentage error that the carpenter can make in calculating the volume of the rectangular box?a)1%b)3%c)10%d)18%e)22%Correct answer is option 'E'. Can you explain this answer?
Question Description
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