The Area of a Sheet

solved Uncertainty

Q. A sheet is measured as 16.2 $a = 16.2$ ± 0.1 $Δ a = 0.1$ cm long and 10.1 $b = 10.1$ ± 0.1 $Δ b = 0.1$ cm wide. Its area is about 163.6 cm². To within how many square centimetres is that known?

Inputs

length

a

16.2

length error

Δ a

0.1

width

b

10.1

width error

Δ b

0.1

Outputs

error

Δ p

How it's solved

Solved with Uncertainty

Given

$a = 16.2$
$Δ a = 0.1$
$b = 10.1$
$Δ b = 0.1$

Find $Δ p$ .

Working

first error over first $δ_{a} = \frac{Δ a}{a}$
$δ_{a} = \frac{0.1}{16.2}$
$δ_{a} = 0.00617283951$
second error over second $δ_{b} = \frac{Δ b}{b}$
$δ_{b} = \frac{0.1}{10.1}$
$δ_{b} = 0.0099009901$
relative errors add $δ_{p} = δ_{a} + δ_{b}$
$δ_{p} = 0.00617283951 + 0.0099009901$
$δ_{p} = 0.0160738296$
product of measurements $p = a \cdot b$
$p = 16.2 \cdot 10.1$
$p = 163.62$
error from relative error $Δ p = δ_{p} \cdot p$
$Δ p = 0.0160738296 \cdot 163.62$
$Δ p = 2.63$

Conclusion

In a product the relative errors add, so the product is less certain than either measurement.

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 :area_of_a_sheet do4  kind :numerical5  describes "A sheet is measured as %{length} ± %{length_error} cm long and %{width} ± %{width_error} cm wide. Its area is " \6          "about 163.6 cm². %{error}?"7  given length: 16.28  given length_error: 0.19  given width: 10.110  given width_error: 0.111  asks error: "To within how many square centimetres is that known"12end

solution.rb

1require_relative "../questions/area_of_a_sheet"2require_relative "../science/units_and_measurement/uncertainty"34solve :area_of_a_sheet, using: Uncertainty do5  take length: :a, length_error: :Delta_a, width: :b, width_error: :Delta_b6  take error: :Delta_p7  answer find_value(:error, within: 0.5)89  conclusion "In a product the relative errors add, so the product is less certain than either " \10             "measurement."11end

playground.rb

1require_relative "../solutions/area_of_a_sheet"23play_with :area_of_a_sheet do4  title "The Area of a Sheet"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