How thick am I?
A closed rectangular box is of uniform thickness x . The box has outer dimensions 6 inches, 4 inches, and 3 inches and it has an inside volume of 30 cubic inches. x can be expressed as b a where a and b are coprime integers, find the value of a + b.
The answer is 3.
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8 solutions
By the way, you can expand the LHS of the equation in line 3 and factorize it again, which is a bit tedious, we have
( 6 − 2 x ) ( 4 − 2 x ) ( 3 − 2 x ) = 3 0
4 2 − 1 0 8 x + 5 2 x 2 − 8 x 3 = 0
8 x 3 − 5 2 x 2 + 1 0 8 x − 4 2 = 0
Now, here's the tricky part. We need to factorize LHS of the equation above, so we have
2 ( 4 x 3 − 2 6 x 2 + 5 4 x − 2 1 ) = 0
Since 2 1 is one of the roots of the equation, using the polynomial long division, we have
2 ( 2 x − 1 ) ( 2 x 2 − 1 2 x + 2 1 ) = 0
But 2 x 2 − 1 2 x + 2 1 has no rational roots, i.e. they have complex roots, so we can stop the factorization process. And, as I mentioned just now, since 2 1 is one of the roots, therefore, x = 2 1 .
From where this 2x came i know 2 is thickness but why 2x why not x
i m afraid...bt from wer did this 2 come? i mean 2x...twice the thickness?
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Since the box is closed on all sides, you have to count from the top and the bottom
Note that the inner volume, in cubic inches, is ( 6 − 2 x ) ( 4 − 2 x ) ( 3 − 2 x ) = 3 0 . Note that x = 2 1 is a solution to this equation, so our answer is 3 .
why can't x = 0.5 = 2/4 then a+b = 6 or x = 0.5 = 3/6 then a+b = 9 and so on... that means no answer to this question
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In most of the questions where the answer is of the form b a , it is mentioned that a and b are co-prime, although it has not been mentioned here.
The dimensions of inner box are 6-2x, 4-2x, 3-2x. Therefore (6-2x) (4-2x) (3-2x) = 30. Solving the equation, we get x=0.5= 1/2. Therefore, a+b=3.
the dimensions of the inner rectangular volume are: (6-2x), (4-2x) and (3-2x) Therefore inner volume is (6-2x).(4-2x).(3-2x) = 30 (given in question) On solving the cubic equation thus obtained, by hit and trial method we get the rational root as 1/2 which fits the "a/b" form mentioned in the problem, since 1 and 2 are co-prime. Thus, a=1, b=2 and answer is 1+2 =3.
(6-2x) x (4-2x) x (3-2x)=30. Just solve for x.
The thickness being uniform, the cuboid inside will have side lengths of (6-2x), (4-2x) and (3-2x) respectively.
The volume is 30 cubic inches, hence we equate the products of the sides to this figure getting
x = 1/2 = a/b. Thus a+b = 3
(6-2x)(4-2x)(3-2x) = 2 3 5. 6-2x>4-2x>3-2x. Therefore by hidden trial equate 6-2x to 5. Resulting in x = 0.5. Putting it in other factors(4-2x and 3-2x) we see that L.H.S = R.H.S hence, x = 1/2 and so a + b = 1+2=3
As we know outer dimensions of box are 6, 4, 3 inches...and thickness of box is x .. So, inner dimensions will be 6-2x , 4-2x, 3-2x
Given that Volume Of Box = 30 cubic inches
This Is Clear That Volume of the box will be only due to inner dimensions of box
So, (6-2x)(4-2x)(3-2x)=30 Solving this we get x= 2/3
So, Answer = 2+1 = 3
Actually, x=1/2. If x was 2/3, the answer would be 2+3=5.
mr minimario is right fella
6 i n ∗ 4 i n ∗ 3 i n = 7 2 i n 3 The inside volume is 3 0 i n 3 so the thickness is not 0 and we have to use a bit of algebra.
( 6 − 2 x ) ( 4 − 2 x ) ( 3 − 2 x ) = 3 0
Unfortunately, I got stuck and used trial and error. I replaced 2 x by 1 and got these brackets:
( 6 − 1 ) ( 4 − 1 ) ( 3 − 1 )
which equalled 30 so I divided 1 by 2 and got an answer of 2 1 . By adding the numerator and the denominator I got an answer of 3.