If $$x +[\frac{1}{(4x)}]=\frac{5}{2}$$, then what is the value of $$\frac{(64x^6 + 1)}{8x^3}$$ ?

To find : $$\frac{(64x^6 + 1)}{8x^3}$$

= $$8x^3+\frac{1}{8x^3}$$

= $$(2x)^3+(\frac{1}{2x})^3$$

Let $$2x=a$$

=> $$x=\frac{a}{2}$$ ---------(i)

Thus, we need to find : $$a^3+\frac{1}{a^3}$$ ----------(ii)

Given : $$x +[\frac{1}{(4x)}]=\frac{5}{2}$$

Substituting value from equation (i),

=> $$\frac{a}{2}+\frac{1}{2a}=\frac{5}{2}$$

=> $$a+\frac{1}{a}=5$$

Cubing both sides, we get :

=> $$(a+\frac{1}{a})^3=(5)^3$$

=> $$a^3+\frac{1}{a^3}+3(a)(\frac{1}{a})(a+\frac{1}{a})=125$$

=> $$a^3+\frac{1}{a^3}+3(5)=125$$

=> $$a^3+\frac{1}{a^3}=125-15=110$$

=> Ans - (A)