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03 Apr 2020, 01:55
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C
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Difficulty:
55%
(hard)
Question Stats:
64% (02:45) correct
36%
(03:05)
wrong
based on 68
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If three diagonals of the faces of a rectangular brick have their diagonals in the ratio \(3 : 2√3 : √15\), then the ratio of the length of the shortest edge of the brick to that of its longest edge is
A. \(√3 : 2\)
B. \(2 : √5\)
C. \(1 : √3\)
D. \(√2 : √3\)
E. \(√2 : √5\)
Are You Up For the Challenge: 700 Level Questions
03 Apr 2020, 06:11
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Let a, b, and c be length of the three sides of the brick.
From the given ratio of the brick's face diagonals, we can calculate as follows:
(1) (a^2+b^2) = (3)^2 = 9
(2) (b^2+c^2) = (2√3)^2 = 12
(3) (a^2+c^2) = (√15)^2 = 15
(1)+(2)+(3)
2*(a^2+b^2+c^2) = 36
(a^2+b^2+c^2) = 18
(9+c^2) = 18
c^2 = 9
(2)
(b^2+c^2) = 12
(b^2+9) = 12
b^2 = 3
(1)
(a^2+b^2) = 9
(a^2+3) = 9
a^2 = 6
From values of a^2, b^2, and c^2, the shortest and the longest sides are obviously b and c, respectively.
b^2 : c^2 = 3 : 9 = 1 : 3
Hence, b : c = 1 : √3
FINAL ANSWER IS (C)
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From the given ratio of the brick's face diagonals, we can calculate as follows:
(1) (a^2+b^2) = (3)^2 = 9
(2) (b^2+c^2) = (2√3)^2 = 12
(3) (a^2+c^2) = (√15)^2 = 15
(1)+(2)+(3)
2*(a^2+b^2+c^2) = 36
(a^2+b^2+c^2) = 18
(9+c^2) = 18
c^2 = 9
(2)
(b^2+c^2) = 12
(b^2+9) = 12
b^2 = 3
(1)
(a^2+b^2) = 9
(a^2+3) = 9
a^2 = 6
From values of a^2, b^2, and c^2, the shortest and the longest sides are obviously b and c, respectively.
b^2 : c^2 = 3 : 9 = 1 : 3
Hence, b : c = 1 : √3
FINAL ANSWER IS (C)
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General Discussion
03 Apr 2020, 02:44
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Quote:
If three diagonals of the faces of a rectangular brick have their diagonals in the ratio 3:2√3:√15, then the ratio of the length of the shortest edge of the brick to that of its longest edge is
A. √3:2
B. 2:√5
C. 1:√3
D. √2:√3
E. √2:√5
A. √3:2
B. 2:√5
C. 1:√3
D. √2:√3
E. √2:√5
Let sides are of length a < b < c
c = ?
3:2√3:√15
i.e. √(b^2+c^2) = √15
i.e. (b^2+c^2) = 15----------(1)
similarly
(a^2+c^2) = (2√3)^2 = 12----------(2)
and
(a^2+b^2) = (3)^2 = 9----------(3)
Subtracting eq(3) from eq (1)
c^2 - a^2 = 15-9 = 6
also (a^2+c^2) = 12
Adding them we get, 2c^2 = 18 i.e. c^2 = 9
i.e. a^2 = 3
Ratio of longest to shortest = c/a = √(c^2/a^2) = √(9/3) = √3
ANswer: Option C
