A Hair Made Large

Q. Under a microscope that makes lengths 100 times larger, a hair looks 3.5 mm wide. How wide is the hair, in metres? expects 3.5 × 10⁻⁵

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The web of concepts

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The equations it used

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Q. Under a microscope that makes lengths 100 $m_{l} = 100$ times larger, a hair looks 3.5 mm $l_{i} = 3.5$ mm wide. How wide is the hair, in metres?

The playground

Inputs

magnification

m_{l}

100

seen width

l_{i}

3.5 mm

Outputs

width

l_{o}

The solution

Solved with Dimensions

Given

$m_{l} = 100$
$l_{i} = 3.5$ mm

Find $l_{o}$ .

Working

$l_{i} = 3.5$ mm $= 0.0035$ m
lengths magnified $m_{l} = \frac{l_{i}}{l_{o}}$
$100 = \frac{0.0035}{l_{o}}$
$l_{o} = 3.5 \times 10^{−5}$ m
$l_{o} \approx 3.5 \times 10^{−5}$ mto 2 significant figures

Conclusion

A magnification multiplies lengths; it does not care what the lengths are of.

The Ruby

1require_relative "../physics"23question :a_hair_made_large do4  kind :numerical5  describes "Under a microscope that makes lengths %{magnification} times larger, a hair looks %{seen_width} wide. %{width}, in metres?"6  given magnification: 100.07  given seen_width: "3.5 mm"8  asks width: "How wide is the hair"9end

1require_relative "../questions/a_hair_made_large"2require_relative "../science/units_and_measurement/dimensions"34solve :a_hair_made_large, using: Dimensions do5  take magnification: :m_l, seen_width: :l_i6  take width: :l_o7  answer find_value(:width, figures: 2)89  conclusion "A magnification multiplies lengths; it does not care what the lengths are of."10end

1require_relative "../solutions/a_hair_made_large"23play_with :a_hair_made_large do4  title "A Hair Made Large"5end

The laws

quantity	written	units	within	from
speed of light	$c$	m/s	—	Units
avogadro constant	$N_{A}$	mol⁻¹	—	Units
length	$l$	m	0 … ∞	Units
mass	$m$	kg	—	Units
time	$t$	s	0 … ∞	Units
electric current	$I$	A	—	Units
temperature	$T$	K	—	Units
amount of substance	$n$	mol	—	Units
luminous intensity	$I_{v}$	cd	—	Units
arc length	$s$	m	—	Units
radius	$r$	m	0 … ∞	Units
plane angle	$θ$	—	—	Units
area	$A$	m²	—	Units
solid angle	$Ω$	—	—	Units
light time	$t_{light}$	s	—	Units
light distance	$d_{light}$	m	—	Units
number of entities	$N$	—	—	Units
baseline	$B$	m	—	Units
distance away	$D$	m	—	Units
parallax angle	$θ_{p}$	—	—	Units
pi	$π$	—	0 … ∞	Dimensions
gravitational constant	$G$	N m² kg⁻²	—	Dimensions
volume	$V$	m³	—	Dimensions
speed	$v$	m/s	—	Dimensions
acceleration	$a$	m/s²	—	Dimensions
force	$F$	N	—	Dimensions
density	$ρ$	kg/m³	—	Dimensions
work	$W$	J	—	Dimensions
surface area	$S$	m²	—	Dimensions
height	$h$	m	—	Dimensions
breadth	$b$	m	—	Dimensions
cylinder surface area	$S_{c}$	m²	—	Dimensions
volume of a slab	$V_{s}$	m³	—	Dimensions
relative density	$ρ_{rel}$	—	—	Dimensions
density of water	$ρ_{w}$	kg/m³	—	Dimensions
linear magnification	$m_{l}$	—	—	Dimensions
image length	$l_{i}$	m	—	Dimensions
object length	$l_{o}$	m	—	Dimensions
image area	$A_{i}$	m²	—	Dimensions
object area	$A_{o}$	m²	—	Dimensions
volume of them all	$V_{N}$	m³	—	Dimensions
molar volume	$V_{m}$	m³	—	Dimensions
room to spare	$R_{V}$	—	—	Dimensions

θ = \frac{s}{r}

arc over radius

Ω = \frac{A}{r^{2}}

area over radius squared

d_{light} = c \cdot t_{light}

distance at light speed

N = n \cdot N_{A}

entities in moles

θ_{p} = \frac{B}{D}

baseline over distance

V = l^{3}

volume of a cube

v = \frac{l}{t}

average speed

a = \frac{v}{t}

average acceleration

F = m \cdot a

newtons second law

ρ = \frac{m}{V}

mass per volume

W = F \cdot l

force through a distance

S = 6 \cdot l^{2}

surface of a cube

V = \frac{4 \cdot π \cdot r^{3}}{3}

volume of a sphere

S_{c} = 2 \cdot π \cdot r \cdot (r + h)

surface of a cylinder

A = l \cdot b

area of a rectangle

V_{s} = l \cdot b \cdot h

volume from three sides

ρ_{rel} = \frac{ρ}{ρ_{w}}

density against water

m_{l} = \frac{l_{i}}{l_{o}}

lengths magnified

{m_{l}}^{2} = \frac{A_{i}}{A_{o}}

areas magnified

V_{N} = N \cdot V

atoms side by side

R_{V} = \frac{V_{m}}{V_{N}}

molar over atomic volume