1.6 Approximation and errorIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   The width of a doorway is recorded as $0.86$ m, correct to 2 decimal places.

   (a)

   What is the smallest possible value of the actual width of the doorway?

   [1 mark]

   • A$0.855$ m
   • B$0.85$ m
   • C$0.859$ m
   • D$0.865$ m

   (b)

   Which inequality gives the range of possible values of the width $w$ metres?

   [1 mark]

   • A$0.85\leq w<0.87$
   • B$0.855<w\leq0.865$
   • C$0.855\leq w\leq0.865$
   • D$0.855\leq w<0.865$

   (c)

   The width of the doorway is later found to be exactly $0.8632$ m. Calculate the percentage error in the recorded width.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A cyclist rides $41.8$ km in $2.3$ hours, with the distance and the time each given to the accuracy shown.

   (a)

   The average speed is found by dividing the distance by the time. Which value gives the speed to an appropriate degree of accuracy, matching the less accurate of the two measurements?

   [1 mark]

   • A$18.17391\ldots$ km h$^{-1}$
   • B$18.2$ km h$^{-1}$
   • C$18$ km h$^{-1}$
   • D$20$ km h$^{-1}$

   (b)

   Which calculation gives the best estimate of the speed, found by rounding each measurement to 1 significant figure?

   [1 mark]

   • A$18.17\ldots$ km h$^{-1}$
   • B$40\div2=20$ km h$^{-1}$
   • C$50\div2=25$ km h$^{-1}$
   • D$40\div2.3\approx17$ km h$^{-1}$

   (c)

   A second student calculates the speed as $1.8$ km h$^{-1}$. Use your estimate to explain why this answer is unreasonable, and suggest the likely error.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A rectangular field is measured as $85$ m by $42$ m, with each measurement correct to the nearest metre.

   (a)

   Find the lower bound and the upper bound of the area of the field.

   [3 marks]

   (b)

   The length and width are later found to be exactly $85.4$ m and $41.7$ m. Calculate the percentage error in the area found using the measured values $85$ m and $42$ m.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A pond is in the shape of a cylinder with radius $12.5$ m, measured correct to 1 decimal place, and depth $1.8$ m, correct to 2 significant figures.

   (a)

   (i) Write down the lower and upper bounds for the radius.
   (ii) Find the lower bound and the upper bound of the area of the surface of the pond.
   (iii) Hence write the area of the surface to an appropriate degree of accuracy, and justify your answer.

   [6 marks]

   (b)

   (i) The radius is later found to be exactly $12.48$ m. Calculate the percentage error in the area found using $r=12.5$ m.
   (ii) Calculate the volume of water in the full pond using the measured values, giving your answer to an appropriate degree of accuracy.

   [6 marks]

   Total for question 4: 12 marks

End of questions