A solid rectangular block has a base which measures $2x$ cm by $x$ cm. The height of the block is $y$ cm and the volume of the block is $72 cm^3$. Express $y$ in terms of $x$ and show that the total surface area, $Acm^2$ , of the block is given by $$A=4x^2+\frac{216}{x}$$
My attempt,
$2x^2y=72$
$y=\frac{36}{x^2}$
$A=2\cdot2x^2+4yx$
$A=4x^2+\frac{144}{x}$
Where have I done mistake?
$\endgroup$ 23 Answers
$\begingroup$The surface area is:
$$A = 2*(2x^2 + 2xy + xy) = 4x^2 + 6xy= 4x^2 + \frac{216}{x}$$
$\endgroup$ $\begingroup$Notice, volume of the rectangular block $$V=\text{(length)}\times \text{(width)}\times \text{(height)}$$ $$(2x)(x)(y)=72$$ $$2x^2y=72$$$$\implies \color{blue}{y=\frac{36}{x^2}}$$ Now, the surface area of the rectangular block $$A=2(\text{(length)}\times \text{(width)}+\text{(length)}\times \text{(height)}+\text{(width)}\times \text{(height)})$$ $$=2(2x\cdot x+2x\cdot y+x\cdot y)$$ $$=4x^2+6xy$$ setting the values of $y$, one should get $$A=4x^2+6x\frac{36}{x^2}$$ $$\bbox[5px, border:2px solid #C0A000]{\color{red}{A=4x^2+\frac{216}{x}}}$$
$\endgroup$ 1 $\begingroup$Total surface area =$2(2x^2+3xy)$ so now $y=36/x^2$ so plugging in values you get $4x^2+\frac{108}{x^2}.2=4x^2+216/x^2$ you have missed $bh,lh$. Hope its clear now.
$\endgroup$
