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Difficulty:
35%
(medium)
Question Stats:
75% (02:20) correct
25%
(03:20)
wrong
based on 51
sessions
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A three-dimensional rectangular box with dimensions X, Y, and Z has faces whose surface areas are 24, 24, 48, 48, 72, and 72 square units. What is X + Y + Z?
A. 18
B. 22
C. 24
D. 30
E. 36
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A. 18
B. 22
C. 24
D. 30
E. 36
_________________
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Bunuel
Answer 22.
Let us consider 24, it can be 3,8 or 4,6. I started my trial and error with 4,6
4*6= 24, okay
6*8=48, okay
But 4*8=32 (Not possible)
However
4*6= 24, okay
4*12=48, okay
6*12= 72. (Correct match)
X:y:z :: 4:6:12
Let 3 sides of cuboid are a,b and c
xy=24, yz=48 and zx=72
x=24/y
zx=72
z*(24/y)=72
z=3y
yz=48 or 3y^2=48
y=4
z=12
and x=6
x+y+z=22
xy=24, yz=48 and zx=72
x=24/y
zx=72
z*(24/y)=72
z=3y
yz=48 or 3y^2=48
y=4
z=12
and x=6
x+y+z=22
Listing down the combinations
XY - 24 - 2*12, 3*8, 4*6
YZ - 48 - 2*14, 3*16, 4*12, 6*8
ZX - 72 - 2*36, 3*24, 4*18, 6*12, 8*9
Assuming z is 9, x is 8, then y = 3 which doesn't validate with YZ.
Using similar logic only sides 12, 4,6 tally with all combinations.
Hence answer is 22
Posted from my mobile device
XY - 24 - 2*12, 3*8, 4*6
YZ - 48 - 2*14, 3*16, 4*12, 6*8
ZX - 72 - 2*36, 3*24, 4*18, 6*12, 8*9
Assuming z is 9, x is 8, then y = 3 which doesn't validate with YZ.
Using similar logic only sides 12, 4,6 tally with all combinations.
Hence answer is 22
Posted from my mobile device
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Bunuel
possible combinations
x*y=24
x*z=48
y*z=72
x=24/y
z*24/x=72
z=3x
3*x^2=48 ; x=4
y=6,z=12
sum ; 22
IMO B
Given,
XY=24--1
XZ=48--2
ZY=72--3
Multiply the above 3 we get
(XYZ)^2=24x48x72=(288)^2
Therefore XYZ=288
Using eqn 1,2 and 3
Z=12
Y=6
X=4
Therefore X+Y+Z=22
Hence B
XY=24--1
XZ=48--2
ZY=72--3
Multiply the above 3 we get
(XYZ)^2=24x48x72=(288)^2
Therefore XYZ=288
Using eqn 1,2 and 3
Z=12
Y=6
X=4
Therefore X+Y+Z=22
Hence B
Bunuel
XY = 24
YZ = 48
XZ = 72
Instead of solving the equations, I will pick the greatest number 72 and look at its similarly sized factors 8 and 9. Since 9 is neither a factor of 24 nor of 48, so the two 3s must be distributed between the two factors. The 8 could be divided as 4 and 2 to give 4*3 and 2*3 i.e. 12 and 6 as factors of 72.
Let's try whether X and Z work as 6 and 12.
Now we see that 6 * 4 = 24 and 12 * 4 = 48 which means that Y = 4.
Hence we get the three dimensions as 4, 6 and 12 and so their sum is 22.
Answer (B)
Check this video for a discussion on rectangular solids: https://www.youtube.com/watch?v=kqVgiZRd6yM
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