If the areas of the adjacent faces of a rectangular block are in the ratio 2 : 3 : 4 and its volume is 9000 cm³ , then the length of the shortest edge is
A. 30 cm
B. 20 cm
C. 15 cm
D. 10 cm
Answers
Given : Areas of three adjacent faces of a rectangular block are in the ratio 2 : 3 : 4 and its volume is 9000 cm³ .
Let the dimensions of a rectangular block be l, b and h .
Let these adjacent faces of a rectangular block are x, y ,z .
x : y : z = 2 : 3 : 4
So, x = lb = 2k , y = bh = 3k , z = hl = 4k and volume of cuboid = V
V = lbh
On squaring both sides :
V² = (lbh)² = l²b²h²
V² = lb × bh × hl
V² = xyz
(9000)² = (2k × 3k × 4k)
81000000 = 24k³
k³ = 81000000/24
k³ = 3375000
k = ³√3375000
k = 150
Now,
lb = 2k
lb = 2 × 150 = 300
bh = 3k
bh = 3 × 150 = 450
hl = 4k
hl = 4 × 150 = 600
Then,
l = lbh/bh = 9000/450
l = 20 cm
b = lbh/lh
b = 9000/600
b = 15 cm
h = lbh/lb
h = 9000/300
h = 30 cm
Hence, the length of the shortest edge is 15 cm .
Option (C) 15 cm is correct.
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Given:
Areas of adjacent faces of a rectangular box are in the ratio = 2 : 3 : 4
Volume = 9000 cu. cm
To find:
Length of the shortest side
Solution:
Let the areas of adjacent sides be = 2x, 3x and 4x
Thus, 2x = lb
3x = bh
4x = lh
Volume = lbh = 9000
Multiply all the above 3 equations:
2x × 3x × 4x = lb × bh × lh
=> 24x^3 = (lbh)^2
=> 24x^3 = (9000)^2
=> x^3 = 3375000
=> x = 150
2x = lb = 300
3x = bh = 450
4x = lh = 600
Length = l = lbh/bh = 9000/450 = 20 cm
Breadth = b = lbh/lh = 9000/600 = 15 cm
Height = h = lbh/lb = 9000/300 = 30 cm