Answer:
(a) The constants required describing the rod's density are B=2.6 and C=1.325.
(b) The mass of the road can be found using [tex]A\int_0^{12}\left(B+Cx)dx[/tex]
Explanation:
(a) Since the density variation is linear and the coordinate x begins at the low-density end of the rod, we have a density given by
[tex]2.6\frac{g}{cm^3}+\frac{18.5\frac{g}{cm^3}-2.6\frac{g}{cm^3}}{12 cm}x = 2.6\frac{g}{cm^3}+1.325x\frac{g}{cm^2}[/tex]
recalling that the coordinate x is measured in centimeters.
(b) The mass of the rod can be found by having into account the density, which is x-dependent, and the volume differential for the rod:
[tex]m=\int\rho dv=\int\left(B+Cx\right)Adx=5\int_0^{12}\left(2.6+1.325x\right)dx=126.6[/tex],
hence, the mass of the rod is 126.6 g.
