A Slide on the Screen

Q. A slide of 1.75 cm² is thrown onto a screen, where its picture covers 1.55 m². How many times are lengths magnified? expects 94.1

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Q. A slide of 1.75 cm² $A_{o} = 1.75$ cm² is thrown onto a screen, where its picture covers 1.55 m² $A_{i} = 1.55$ m². How many times are lengths magnified?

The playground

Inputs

slide area

A_{o}

1.75 cm²

picture area

A_{i}

1.55 m²

Outputs

magnification

m_{l}

The solution

Solved with Dimensions

Given

$A_{o} = 1.75$ cm²
$A_{i} = 1.55$ m²

Find $m_{l}$ .

Working

$A_{o} = 1.75$ cm² $= 1.75 \times 10^{−4}$ m²
areas magnified ${m_{l}}^{2} = \frac{A_{i}}{A_{o}}$
${m_{l}}^{2} = \frac{1.55}{1.75 \times 10^{−4}}$
$m_{l} = −94.1123948, 94.1123948$ the positive one is taken
$m_{l} = 94.1123948$
$m_{l} \approx 94.1$ to 3 significant figures

Conclusion

Areas grow as the square of lengths, so the linear magnification is the square root of the areas' ratio.

The Ruby

1require_relative "../physics"23question :a_slide_on_the_screen do4  kind :numerical5  describes "A slide of %{slide_area} is thrown onto a screen, where its picture covers %{picture_area}. %{magnification}?"6  given slide_area: "1.75 cm2"7  given picture_area: "1.55 m2"8  asks magnification: "How many times are lengths magnified"9end

1require_relative "../questions/a_slide_on_the_screen"2require_relative "../science/units_and_measurement/dimensions"34solve :a_slide_on_the_screen, using: Dimensions do5  take slide_area: :A_o, picture_area: :A_i6  take magnification: :m_l7  answer find_value(:magnification, figures: 3, root: :positive)89  conclusion "Areas grow as the square of lengths, so the linear magnification is the square root " \10             "of the areas' ratio."11end

1require_relative "../solutions/a_slide_on_the_screen"23play_with :a_slide_on_the_screen do4  title "A Slide on the Screen"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