The average density of a carbon-fiber-epoxy composite is 1.615 g/cm3. the density of the epoxy resin is 1.21 g/cm3 and that of the carbon fibers is 1.74 g/cm3. (a) what is the volume percentage of carbon fibers in the composite? (b) what are the weight percentages of epoxy resin and carbon fibers in the composite?

The composite material is composed of carbon fiber and epoxy resins. Now, density is an intensive unit. So, to approach this problem, let's assume there is 1 gram of composite material. Thus, mass carbon + mass epoxy = 1 g.

Volume of composite material = 1 g / 1.615 g/cm³ = 0.619 cm³
Volume of carbon fibers = x g / 1.74 g/cm³ 
Volume of epoxy resin = (1 - x) g / 1.21 g/cm³ 

a.) V of composite = V of carbon fibers + V of epoxy resin
0.619 = x/1.74 + (1-x)/1.21
Solve for x,
x = 0.824 g carbon fibers
1-x = 0.176 g epoxy resins

Vol % of carbon fibers = [(0.824/1.74) ÷ 0.619]*100 = 76.5%

b.) Weight % of epoxy = 0.176 g epoxy/1 g composite * 100 = 17.6%
     Weight % of carbon fibers = 0.824 g carbon/1 g composite * 100 = 82.4%