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If $x = \frac{y}{2}$ ,evaluate\(\left(\frac{x^{3}}{y^{3}...

If $x = \frac{y}{2}$ ,evaluate $\left(\frac{x^{3}}{y^{3}}+\frac{1}{2}\right) \div \left(\frac{1}{2} - \frac{x^{2}}{y^{2}}\right)$

A.

5/8

B.

5/2

C.

5/32

D.

5/16

View answer & explanation

Correct answer is B

$x = \frac{y}{2}$

$\left(\frac{x^{3}}{y^{3}}+\frac{1}{2}\right) \div \left(\frac{1}{2} - \frac{x^{2}}{y^{2}}\right)$

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

= $\frac{y^{3}}{8} \times \frac{1}{y^3} + \frac{1}{2}$

= $\frac{1}{8} + \frac{1}{2}$

= $\frac{5}{8}$

$\frac{1}{2} - \frac{x^2}{y^2} = \frac{1}{2} - (\frac{y}{2})^{2} \div y^2)$

= $\frac{1}{2} - \frac{y^2}{4} \times \frac{1}{y^2}$

= $\frac{1}{2} - \frac{1}{4}$

= $\frac{1}{4}$

$\therefore \left(\frac{x^{3}}{y^{3}}+\frac{1}{2}\right) \div \left(\frac{1}{2} - \frac{x^{2}}{y^{2}}\right) = \frac{5}{8} \div \frac{1}{4}$

= $\frac{5}{2}$

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