The volume of a rectangular parallel pipe is given as 144 cc. Its surface area is given as 192 sq cm. And its corner to corner diagonal is given as 13 cm. How do I find out the three sides?
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Answers
The pipe is clearly a cuboid .
Let the three sides be l , b and h
Volume of cuboid
We know that volume = l b h
Given: V = 144 c³
==> l b h = 144 c³=========>(1)
Surface area of cuboid :
We know that surface area of cuboid = 2 ( lb + lh + bh )
Given S = 192 c²
==> 2 ( lb + bh + lh ) = 192 c²
==> lb + bh + lh = 96 c²======>(2)
Diagonal
We know that diagonal of cuboid = √( l² + b² + h² )
Given diagonal = 13 cm
==> √( l² + b² + h² ) = 13 c
Squaring both sides we get:
==> l² + b² + h² = 169 c²=======> (3)
Calculating the sides
Multiply (2) by 2:
2( lb + bh + lh ) = 192 c² .......................(4)
From (3)
l² + b² + h² = 169 c² ..........................(5)
Add (4) and (5)
==> l² + b² + h² + 2 ( lb +lh + bh ) = 192 c² + 169 c²
==> ( l + b + h )² = 361 c²
==> l + b + h = 19 c ...........................(6)
Assuming that l,b and h are integers
The least value of l,b and h can be 1.
l + b + h = 19
∴ l < 18
b < 18
h < 18
l , b , h ∈ { 1,2,3...................17}
Also :
l² + b² + h² = 169
Now the only way for me is to trial and error !
Trial and error method ( my favourite )
l² + b² + h² = 169
Only possibility if l,b and h are integers :
169 as sum of 2 squares trial and error this .
x² + y² = 169
Only 5² + 12² = 169 [ no other value found ]
Hence either 12² or 5² should be the sum of squares now.
By trial and error :
3² + 4² = 5² [ no other value found for 12² ]
Hence the value of l , b and h has to be 3 , 4 , and 5
3² + 4² + 12² = 169
= 9 + 16 + 144 = 169
= 169 = 169 This satisfies the equation too.
Answer
The only possible values of l , b and h are
3 , 4 and 12 cm
# JISHNU
Hope it helps :)
Note that if l , b and h are not integers , then many solutions are possible !
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Answer:
Step-by-step explanation:
