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#?

2 Answers
Sep 5, 2016

Answer:

Yes, the tea leaf sets off the alarm.

Explanation:

The question is asking you to use density, #rho#, which is #"mass"/"volume"#, and usually has units of #g*cm^-3# or #g*mL^-1# in order to determine the mass of each substance.

Since #"density, "rho# #=# #"Mass"/"Volume"#, #"mass"="volume"xxrho#. The volume of a cylinder is #pi*r^2*l#, where #"l"# is cylinder length.

I need to find (i) the volume of the cylinder #=# #pixx"radius"^2xx"length"#, and (ii) the mass of the substance it contains, #=# #rhoxx"volume"#.

#"Mass of gold"# #=# #19.3*g*cancel(cm^-3)xxpixx3.8^2*cancel(cm^2)xx22*cancel(cm)#

#=# #19252*g#.

#"Mass of sand"# #=# #3.0*g*cancel(cm^-3)xxpixx3.8^2*cancel(cm^2)xx22*cancel(cm)#

#=# #2993*g#.

Given that gold is currently selling for #$1326*USD*"oz"^-1# (#1*"oz"=28.5*g#), how much do you think that gold bar is worth?

Just to add that a #"bag of sand"# is cockney rhyming slang for a #£1000-00#, and this amused me....

Sep 5, 2016

Answer:

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_"cylinder"=pi*r^2*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 #"g/cm"^3"#, I'm going to use centimeters.

#V_"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.

#"density"=("mass")/("volume")"#

We can rearrange the equation to determine mass.

#"mass"="density"xx"volume"#

Determine the mass of sand.

#"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.

#"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.