A covered box of volume  72 cm3 and the  base sides in a ratio of 1:2 is to be made. The length of all sides so that the total surface area is the least possible is

# A covered box of volume  72 ${\mathrm{cm}}^{3}$ and the  base sides in a ratio of 1:2 is to be made. The length of all sides so that the total surface area is the least possible is

1. A

2, 4, 9

2. B

8, 3, 3

3. C

6, 6, 2

4. D

6, 3, 4

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### Solution:

Let  be the dimensions , so $V=lbh=2{b}^{2}h$

The surface area

$=2\left(2{b}^{2}+\frac{3b}{b}+2b×\frac{36}{{b}^{2}}\right)$

$=2\left(2{b}^{2}+\frac{108}{b}\right)$

$=4\left({b}^{2}+\frac{54}{b}\right)$

$\frac{dS}{db}=4\left(2b-\frac{54}{{b}^{2}}\right),\frac{dS}{db}$ is zero is $b=3$ and

$\frac{{d}^{2}S}{d{b}^{2}}=4\left(2+\frac{108}{{b}^{3}}\right)>0$

Hence $S$ is minimum when $b$ = 3. So the dimensions are

$6,3,\frac{36}{9}=6,3,4$