Answer:
The density of the ring is:
[tex]\rho=17.5\pm 3 \, g/cm^3[/tex]
This means the ring could very well be made of gold, but it is very unlikely that it is made of brass.
Explanation:
For a quantity f(x,y) that depends on other quantities (in this case two) x and y, the error is given by:
[tex]\sigma_f=\sqrt{\left(\frac{\partial f}{\partial x}\right)^2\sigma_x^2+\left(\frac{\partial f}{\partial y}\right)^2\sigma_y^2 }[/tex]
where [tex]\sigma_x[/tex] and [tex]\sigma_y[/tex] are the standard deviations on errors of the variables [tex]x[/tex] and [tex]y[/tex].
In our case [tex]\rho=f(m,V)=\frac{m}{V}[/tex] where [tex]m[/tex] is the mass and [tex]V[/tex] is the volume.
Knowing that [tex]\sigma_m=0.02[/tex] and [tex]\sigma_V=0.03[/tex] we can estimate the error on the density
[tex]\sigma_{\rho}=\sqrt{\left(\frac{1}{V}\left)^2\sigma_m^2+\left(\frac{m}{V^2}\right)^2\sigma_V^2} [/tex][tex]\approx 3[/tex] (values were directly plugged)
The density is by using the given values
[tex]\rho=\frac{m}{V}=\frac{2.80}{0.16}=17.5 \, g/cm^3[/tex]
The density with error is given by
[tex]\rho=\frac{m}{V}\pm \sigma_{\rho}=17.5\pm 3 \, g/cm^3[/tex]
Which means it could go as high as 20.5 or as low as 14.5, Meaning that the ring could very well be made of gold, but it is very unlikely that it is made of brass.
