A rectangular solid has a surface area of 94 square inches. What is it

Great (and tricky) question!

Target question: What is the volume of the rectangular solid?

This is a good candidate for rephrasing the target question.

Let x, y, and z be the dimensions (length, width and height) of the rectangular solid.
So, xyz is its volume

REPHRASED target question: What is the value of xyz?

Given: The rectangular solid has a surface area of 94 square inches
Each side of the rectangular solid is a RECTANGLE.
So, 2 sides have the dimensions x by y, 2 sides have the dimensions x by z, and 2 sides have the dimensions y by z
So, the total surface area = 2(xy + xz + yz) = 94
Divide both sides by 2 to get: xy + xz + yz = 47

Statement 1: The surface area of one side of the rectangular solid is 12.
Let's say this is the side with dimensions x by y
So, xy = 12
Is this enough information to determine the value of xyz? No.
We know nothing about the value of z.
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The surface area of one side of the rectangular solid is 20
Let's say this is the side with dimensions x by z
So, xz = 20
For the same reason as above, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that xy = 12
Statement 2 tells us that xz = 20
We also know that xy + xz + yz = 47

NOTE: since this is system of 3 equations with 3 variables, many people will conclude that we can solve it for x, y, and z. However, that rule only applies to linear equations. In this case, the equations are quadratic equations. So, let's see if we can solve this system and find the value of xyz

Take xy + xz + yz = 47 and replace xy with 12 and xz with 20
We get: 12 + 20 + yz = 47
So, yz = 15

We now have:
xy = 12, xz = 20 and yz = 15
This means that (xy)(xz)(yz) = (12)(20)(15)
Simplify: x²y²z² = 3600
Rewrite as: (xyz)² = 3600
So, xyz = 60....DONE!!!
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer:

ASIDE: Some people will say that, if (xyz)² = 3600, then EITHER xyz = 60 OR xyz = -60
However, we're dealing with a real world problem here, and in the real world, the volume cannot be negative. So, we can ignore the negative solution.


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2(lb+BH+lh)=94
I.e. lb+BH+lh=47

Statement 1: lb=12
Not sufficient

Statement 2: bh=20
Not sufficient

Combining
lh= 47-12-20=15

Only possibility of l,b,h are 3,4,5
Hence volume = 60

Sufficient

Answer Option C
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I have a different answer to this. I find each of the information suffecient. Please hear me out

Let the sides be a, b and c.

2(ab+bc+ac) = 94
Therefore, ab+bc+ac = 47.

Now, statement 1 = lets assume ab, side is 12

therefore, bc+ac = 35
(b+a).c =35
Now either b+a = 7 and c = 5 or
b+a = 5, c = 7
If b+a= 7, possible values of abc are (5,2,5) (4,3,5) or (6,1,5). However, only 1 combination gives 1 side area as 12 which is the (4,3,5)

If b+a=5 possible values of abc are (3,2,7) ,(4,1,7) . Ans none of them give 12 as the area. b+a will not be 5.

Therefore, the only possible value is (3,4,5)

Similarly, I found the same value by Statement 2 and I think both are sufficient on their own.

Please help me understand where I am wrong.

Bunuel VeritasPrepKarishma
Could you please clarify, as even I feel that answer should be D.

l*b+b*w+w*l=47

From B: Assuming l*b=20, we have w(l+b)=27, only combination that satisfies is l & b equals 5 & 4, hence w=3.
Similarly answer can be derived from A as well.

We are not told that the dimensions of the solid are necessarily integers.
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1) insufficient LB =12 H=?
2) insufficient BH = 20 L=?
combine 1&2
LB=12, BH=20
total area =94
2(LB+LB+LH) =94
LB+BH+LH=47
LB=12 BH=20 so LH=15
we dont require to calculate l,b,h
JUST MULTIPLY ALL
LBxBHxLH= (LBH)^2 = V^2
12x20x15 = V
Definite Ans C

Thanks Bunuel and Sahilvijay

Answer is C

We are given volume as 94
Let's assume a,b,c as the 3 sides. It implies that abc=94

Statement 1: one side area is 12
Let these two sides be a and b. Thus, a*b= 12
Statement 2: other area is 20.
Thus, let these sides be b*c=20(wouldn't make a difference if you choose a*c)

Now we have 3 equations and 3 variables. Which will help us fetch the 3 sides. Thus, we will have all the values to use the surface area formula, 2(ab+bc+ac)

Note: please don't waste time finding these variables. 3 unique equations will give you the values of 3 variables.

Sent from my DUK-L09 using GMAT Club Forum mobile app

GMATinsight,

Very nice explanation, But I want to add one thing...
After this step "lh= 47-12-20=15"

we need not find the value of l, b, h .
Volume = lbh = (lb*bh*hl)^0.5 and hence it can be calcualted to be 60 (thought we need not calculate).

Hence C is the answer.

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