Question

if the area of the adjacent faces of a rectangular block are in the ratio 2:3:4 and its volume is 9000 cm^3, find the length of the shortest side.

Answer 1

Step-by-step explanation:

Let, the edges of the cuboid be a cm, b cm and c cm. The areas of the three adjacent faces are in the ratio 2 : 3 : 4. Thus, length of the shortest edge is 15 cm .

Answer 2

$\star\red {Given:-}$

$Areas \: \: of \: \: three \: \: adjacent \: \: faces \: \: of \: \: a \: \: \\ rectangular \: \: block \: \: are \: \: in \: \: the \: \: ratio \: \: 2 : \\ 3:4 \: \: and \: \: its \: \: volume \: \: is \: \: 9000 \: \: cm ^{2} .$

$Let \: \: the \: \: dimensions \: \: of \: \: a \: \: rectangular \: \: \\ block \: \: be \: \: l \: \: , \: \: b \: \: and \: \: h. \\ \\ Let \: \: these \: \: adjacent \: \: faces \: \: of \: \: a \: \: rect - \\ angular \: \: block \: \: are \: \: x \: , \: y \: , \: z. \: \: \\ \\x \: : \: y \: : \: \: z \: : \: \: = \: 2 \: \: : 3 \: : \: 4 \\ \\ So , x = lb = 2k, y = bh = \: 3k \: , \: z \: = \: hl \: \\ = 4 \: k \: and \: \: volume \: \: of \: \: cuboid \: = V \\ \\ V = \: lbh$

$\pink{On \: \: squaring \: \: both \: \: sides \: : }$

$↦ \: {v}^{2} \: = \: ( \: lbh \: )^{2} = {l}^{2} \: \: {b}^{2} \: \: {h}^{2} \:$

$↦ \: {v}^{2} \: = \: lb \: \times \: bh \: \times \: hl\:$

$↦ \: {v}^{2} \: = \: x \: y \: z$

$→ \: \: ( \: 9000 \: ) \: ^{2} = \: (2k \: \times \: 3k \: \times \: 4k) \:$

$→ \: 31000000 \: = \: 24 \: k \: ^ 3 \:$

$→k \: ^ 3 \: = \: \frac{8100000\:}{ \: 24}$

$→k \: ^ 3 \: = \: 337500$

$→k \: ^ 3 \: = \: 3\sqrt{337500}$

$\blue{→k \: \: = \:150}$

$\green{Now, } \\ \\ lb = 2k \\ lb = 2 \: \times \: 150 = 300 \\ \\ bh = 3k \\ bh = 3 \: \times \: 150 = 450 \\ \\ hl= 4k \\ hl= 4 x 150 = 600 \\ \\ \\ \:$

$\blue {Then, } \\ \\ l = \frac{lbh}{bh} = \frac{9000}{450 1 \: } \\ l = 20 cm \\ b = \frac{lbh}{bh} \: \: \\ b= \: \: \frac{9000}{ 600} \\ b = 15 cm \\ h = \frac{lbh}{bh} \: \\h= \frac{9000}{ 3 \: 00} \: \\ h= 30 cm \:$

$\pink{Hence, \: the \: length \: of \: the \: s hortest \: edge \: is \: 15 cm \: }$

$\huge \red {hope \: it \: hleps \: u}$