Question

The linear density of a rod of length 4 m is given by (x) = 2/sqrt(x), in grams per centimeter, where x is measured in centimeters from one end of the rod. Find the mass of the rod.

Answer 1

The density of a substance is the ratio of its mass and its volume. The mass of the rod is 10,666.67 grams or 10.67 kgs.

What is the density of a substance?

The density of a substance is the ratio of its mass and its volume. Therefore we can write,

$d = \dfrac{m}{v}$

mass of substance = m kg

density of substance = d kg/m³

volume of that substance = v m³

The length of the rod is 4 meters, therefore, the length of the rod in centimeters will be 400 centimeters. Thus, The mass in terms of density can be written as,

$m = \int_0^4(2\sqrt x)dx$

     $= [2\cdot \dfrac{2x^{\frac32}}{3}]_0^{400}\\\\= [\dfrac{4x^{\frac32}}{3}]_0^{400}\\\\=[\dfrac{4(400)^{\frac32}}{3}]-[\dfrac{4(0)^{\frac32}}{3}]\\\\= 10,666.\bar6\rm\ grams$

Hence, the mass of the rod is 10,666.67 grams or 10.67 kgs.

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