# Given rho_"gold"=19.3*g*cm^-3, will a thief set off a mass-activated alarm if he steals a gold cylinder which is 22*cm long, and which has a diameter of 7.6*cm, and replaces the gold bar with a bag of sand whose mass is 3*kg?

Sep 5, 2016

Yes, the tea leaf sets off the alarm.

#### Explanation:

The question is asking you to use density, $\rho$, which is $\text{mass"/"volume}$, and usually has units of $g \cdot c {m}^{-} 3$ or $g \cdot m {L}^{-} 1$ in order to determine the mass of each substance.

Since $\text{density, } \rho$ $=$ $\text{Mass"/"Volume}$, $\text{mass"="volume} \times \rho$. The volume of a cylinder is $\pi \cdot {r}^{2} \cdot l$, where $\text{l}$ is cylinder length.

I need to find (i) the volume of the cylinder $=$ $\pi \times \text{radius"^2xx"length}$, and (ii) the mass of the substance it contains, $=$ $\rho \times \text{volume}$.

$\text{Mass of gold}$ $=$ $19.3 \cdot g \cdot \cancel{c {m}^{-} 3} \times \pi \times {3.8}^{2} \cdot \cancel{c {m}^{2}} \times 22 \cdot \cancel{c m}$

$=$ $19252 \cdot g$.

$\text{Mass of sand}$ $=$ $3.0 \cdot g \cdot \cancel{c {m}^{-} 3} \times \pi \times {3.8}^{2} \cdot \cancel{c {m}^{2}} \times 22 \cdot \cancel{c m}$

$=$ $2993 \cdot g$.

Given that gold is currently selling for $1326*USD*"oz"^-1 ($1 \cdot \text{oz} = 28.5 \cdot g$), how much do you think that gold bar is worth? Just to add that a $\text{bag of sand}$is cockney rhyming slang for a £1000-00, and this amused me.... Sep 5, 2016 Refer to the explanation. #### Explanation: First determine the volume of the cylinders. Then use the densities of the sand and gold to calculate their masses. ${V}_{\text{cylinder}} = \pi \cdot {r}^{2} \cdot h$, where $\pi$is pi (I'm going to use the $\pi$button on my calculator), $r$is the radius, and $h$is the height. Since no units for the radius and height (length) are given, but your densities are given in $\text{g/cm"^3}$, I'm going to use centimeters. ${V}_{\text{cylinder"=pi*(3.8 "cm")^2*22 "cm"="998 cm"^3}}$Now we'll use the volume of the cylinders and the densities of sand and gold to determine their masses. $\text{density"=("mass")/("volume")}$We can rearrange the equation to determine mass. $\text{mass"="density"xx"volume}$Determine the mass of sand. $\text{mass"_"sand"=3.00 "g"/cancel"cm"^3xx998 cancel"cm"^3=3.0xx10^3 "g}$(rounded to two significant figures) Determine the mass of gold. $\text{mass"_"gold"=19.3 "g"/cancel"cm"^3xx998 cancel"cm"^3=19000 "g}\$ (rounded to two significant figures)

As you can see, the sand's mass is much lower than that of the gold, so the alarm would be expected to go off.