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Helen draws a random circle. she then measure it diameter and circumference. should get a circumference, C, of 405mm correct to 3 significant figures she get a diameter, D, of 130mm correct 2 significant figures. Helen want to find the value of Pi using a formula Pi =D/C calculate the lower bound and upper bound for Helen's value of pi give your answer correct to 3 decimal place?

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Mekha Sanjeev

Engineering PhD student

First, you want to find the lower and upper bounds of the circumference C and diameter D. C = 405mm to 3sf so the bounds are: 404.5 to 405.5 We find this by subtracting/adding half the degree of accuracy. Here it is 1mm, looking at 3rd significant figure. D = 130mm to 2sf so the bounds are: 125 to 135 Here the degree of accuracy is 10mm, looking at the 2nd significant figure. Now you consider the combination of figures which will give you the lowest and highest possible value for pi, as given by pi = C/D To get the lowest value we can see that we need to combine the smallest possible C with the highest possible D LB: 404.5/135 = 2.996 (3dp) For the highest value we need the reverse UB: 405.5/125 = 3.224 (3dp) Notice how the true value of pi falls within these bounds.

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Mujavid Bukhari

Software Engineer, Ex-Googler and Computer Science at UCL

If we’re looking for the lowerbound of pi, the circumference C would have to be at its smallest and the diameter D would have to be at its highest. For the upper bound it would be the other way around. So for the lower bound: 405mm (3 s.f.) => 404.5 (lowest possible) 130mm (2 s.f.) => 134.499... (highest possible) lower bound pi = 3.007 (3d.p) So for the higher bound: 405mm (3 s.f.) => 405.4999… (highest possible) 130mm (2 s.f.) => 125.0 (lowest possible) upper bound pi = 3.244 (3d.p)

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