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A rectangular box with a cover is to have a square base. The volume is to be 10 cubic $\mathrm{cm}$. The surface area of the box in terms of the side $x$ is given by which one of the following functions? $\quad$
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Verified Answer
The correct answer is:
$f(x)=(40 / x)+2 x^{2}$
Let the height of rectangular box be $y \mathrm{~cm}$. $\therefore$ Volume $=x \times x \times y$
$\Rightarrow y=\frac{10}{x^{2}}$
Now, surface area of box $=2\left(x^{2}+x y+y x\right)=2\left(x^{2}+2 x y\right)$
$=2\left(x^{2}+\frac{20}{x}\right) \quad$ From equation (i)
$=2 x^{2}+\frac{40}{x}$
$\Rightarrow y=\frac{10}{x^{2}}$
Now, surface area of box $=2\left(x^{2}+x y+y x\right)=2\left(x^{2}+2 x y\right)$
$=2\left(x^{2}+\frac{20}{x}\right) \quad$ From equation (i)
$=2 x^{2}+\frac{40}{x}$
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