Error Bounds and Approximations
You're viewing free preview questions — 4 questions previewing here — subscribe to unlock 51 more, plus the Full Question Bank, Mock Papers & Worksheets.Upgrade

Error Bounds and Approximations — Free MYP4 Mathematics (Standard) Practice Questions

1QuestionUnderstanding Absolute and Percentage ErrorAssessment Practice
4 marks~6 minCriterion D
Biologists use the capture-recapture method to estimate the population size of fish in a lake. They capture, mark, and release 50 fish. One week later, they capture 80 fish and find that 10 are marked. The estimated population size NN is given by:

N=number marked in first catch×total in second catchnumber marked in second catchN = \frac{\text{number marked in first catch} \times \text{total in second catch}}{\text{number marked in second catch}}
a
Calculate NN. [1]
b
State two assumptions the model requires for NN to be a reliable estimate. [1]
c
A biologist suspects that marked fish are more visible to predators, reducing their survival rate before the second catch. Analyse how this would affect the value of NN calculated using the formula, and justify whether the estimate of 400 fish should be considered an overestimate or underestimate of the true population. [2]
Question diagram

Solutions

2QuestionUnderstanding Absolute and Percentage ErrorAssessment Practice
8 marks~12 minCriterion A
A manufacturing engineer measures two components of a mechanical assembly. Component a=5.0±0.1a = 5.0 \pm 0.1 cm and component b=3.0±0.1b = 3.0 \pm 0.1 cm.

Use: absolute error of a product =a0 Δb+b0 Δa+Δa Δb= a_0\,\Delta b + b_0\,\Delta a + \Delta a\,\Delta b, where a0a_0, b0b_0 are measured values and Δa\Delta a, Δb\Delta b are absolute errors.
a
Calculate the absolute error and percentage error for the sum a+ba + b and the product a×ba \times b. [2]
b
A second engineer measures c=6.0±0.2c = 6.0 \pm 0.2 cm and d=4.0±0.2d = 4.0 \pm 0.2 cm. Deduce whether the percentage error in the product c×dc \times d is greater than, equal to, or less than the percentage error in a×ba \times b, and explain why. [3]
c
Prove algebraically that when two measured quantities are added, the maximum absolute error of the sum equals the sum of the individual absolute errors. Then advise the quality control team whether the assembly, with total length a+ba + b, should be approved for a specification requiring the total to lie within ±0.15\pm 0.15 cm of 8.0 cm. [3]
Question diagram

Solutions

3QuestionCalculations Using Rounded ValuesAssessment Practice
2 marks~3 minCriterion B
When two numbers are each rounded to nn decimal places, the sum is also expressed to nn decimal places.

The four sums below illustrate this:

3+4=73 + 4 = 7 (rounded to 0 decimal places)
3.2+4.3=7.53.2 + 4.3 = 7.5 (rounded to 1 decimal place)
3.25+4.31=7.563.25 + 4.31 = 7.56 (rounded to 2 decimal places)
3.253+4.314=7.5673.253 + 4.314 = 7.567 (rounded to 3 decimal places)

Analyse the relationship between the number of decimal places in the addends and in their sum. Hence, formulate a conjecture for the maximum possible error in the sum when both addends are rounded to nn decimal places. [2]

Solutions

4QuestionUnderstanding Absolute and Percentage ErrorAssessment Practice
4 marks~6 minCriterion C
A student measures the time for a ball to fall from a fixed height, repeating the experiment four times. The accepted value, determined by a precision instrument, is 3.03.0 seconds.

Trial 1: 3.03.0 s
Trial 2: 3.23.2 s
Trial 3: 2.82.8 s
Trial 4: 3.13.1 s

Percentage error=∣Approximate value−Exact value∣∣Exact value∣×100%\text{Percentage error} = \frac{|\text{Approximate value} - \text{Exact value}|}{|\text{Exact value}|} \times 100\%
a
Calculate the percentage error for Trials 2, 3, and 4. [2]
b
The student claims their method is reliable because the mean of all four trials equals the accepted value. Calculate the mean of the four trials, then justify whether this claim is valid. [2]

Solutions

Questions 5-55

Unlock these questions with Premium

Sign Up →

That's every free question here — 4 of 55.

+51 more questions, plus Mock Papers & Worksheets, are waiting in Premium.

Upgrade