A Box and Two Nuggets

solved Uncertainty

Q. A box weighs 2.3 kg $a = 2.3$ kg. Two gold nuggets of 20.15 g and 20.17 g, together 0.04032 kg $b = 0.04032$ kg, are put into it. What does the box weigh now, in kilograms, to the right figures?

Inputs

box

a

2.3 kg

nuggets

b

0.04032 kg

Outputs

weight

t

How it's solved

Solved with Uncertainty

Given

$a = 2.3$ kg
$b = 0.04032$ kg

Find $t$ .

Working

sum of measurements $t = a + b$
$t = 2.3 + 0.04032$
$t = 2.34032$
$t \approx 2.3$ to 2 significant figures

Conclusion

A sum can be no more exact than its roughest part. It keeps the fewest decimal places of what went into it, and here the gold is lost in the roughness of the box's weighing.

The web of concepts

given read from the words equation asked

The Ruby that solves it

The question and its solution are small Ruby files. The engine runs them.

question.rb

1require_relative "../physics"23question :a_box_and_two_nuggets do4  kind :numerical5  describes "A box weighs %{box}. Two gold nuggets of 20.15 g and 20.17 g, together %{nuggets}, are " \6          "put into it. %{weight}, in kilograms, to the right figures?"7  given box: "2.3 kg"8  given nuggets: "0.04032 kg"9  asks weight: "What does the box weigh now"10end

solution.rb

1require_relative "../questions/a_box_and_two_nuggets"2require_relative "../science/units_and_measurement/uncertainty"34solve :a_box_and_two_nuggets, using: Uncertainty do5  take box: :a, nuggets: :b6  take weight: :t7  answer find_value(:weight, figures: 2)89  conclusion "A sum can be no more exact than its roughest part. It keeps the fewest decimal " \10             "places of what went into it, and here the gold is lost in the roughness of the box's " \11             "weighing."12end

playground.rb

1require_relative "../solutions/a_box_and_two_nuggets"23play_with :a_box_and_two_nuggets do4  title "A Box and Two Nuggets"5end

The laws it drew on

quantity	written	units	within	from
first measured	$a$	—	—	Uncertainty
error in first	$Δ a$	—	—	Uncertainty
relative error in first	$δ_{a}$	—	—	Uncertainty
second measured	$b$	—	—	Uncertainty
error in second	$Δ b$	—	—	Uncertainty
relative error in second	$δ_{b}$	—	—	Uncertainty
product	$p$	—	—	Uncertainty
error in product	$Δ p$	—	—	Uncertainty
relative error in product	$δ_{p}$	—	—	Uncertainty
total	$t$	—	—	Uncertainty
difference	$d$	—	—	Uncertainty
least count	$l_{c}$	m	—	Uncertainty
scale division	$d_{m}$	m	—	Uncertainty
divisions read past	$N_{d}$	—	—	Uncertainty

δ_{a} = \frac{Δ a}{a}

first error over first

δ_{b} = \frac{Δ b}{b}

second error over second

p = a \cdot b

product of measurements

δ_{p} = δ_{a} + δ_{b}

relative errors add

Δ p = δ_{p} \cdot p

error from relative error

t = a + b

sum of measurements

d = a - b

difference of measurements

l_{c} = \frac{d_{m}}{N_{d}}

division shared out