A polyimide matrix with an elastic modulus of $0\mathrm{.}3{10}^6$ is to be reinforced with $74$vol$\%$ carbon fibers to give a minimum modulus of elasticity of $40{10}^6.$ Determine the Elastic modulus of the carbon fibers.

a$)49\mathrm{.}356{10}^6$

b$)53\mathrm{.}957{10}^6$

c$)33\mathrm{.}227{10}^6$

d$)95\mathrm{.}727{10}^6$

Hint

$E_g=E_f{V}_f+E_m{V}_f$

$E-$ Elastic modulus; $V-$Volume fraction

Subscripts: $f-$ fiber; $m-$ matrix