The Area of a Sheet

Q. A sheet is measured as 16.2 ± 0.1 cm long and 10.1 ± 0.1 cm wide. Its area is about 163.6 cm². To within how many square centimetres is that known? expects 2.63

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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?

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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

The solution

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 Ruby

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

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

1require_relative "../solutions/area_of_a_sheet"23play_with :area_of_a_sheet do4  title "The Area of a Sheet"5end

The laws

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