The length, width, and height of a rectangular box

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The length, width, and height of a rectangular box, in centimeters, are L, W, and H. If the volume of this box is V cubic centimeters and the total area of the 6 sides of this box is A square centimeters, what is the value of V/A ?

(1) At least 2 of L, W, and H are equal to 5.
(2) L, W, and H all have the same value.

C

Source: Official Guide 2020

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by [email protected] » Thu May 02, 2019 9:56 pm

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AbeNeedsAnswers wrote:The length, width, and height of a rectangular box, in centimeters, are L, W, and H. If the volume of this box is V cubic centimeters and the total area of the 6 sides of this box is A square centimeters, what is the value of V/A ?

(1) At least 2 of L, W, and H are equal to 5.
(2) L, W, and H all have the same value.

C

Source: Official Guide 2020
With the given information, we have

Volume of the box = V = LWH; and
Area of the box = A = 2(LW + WH + LH)

Thus, V/A = LWH / 2(LW + WH + LH)

Let's take each statement one by one.

(1) At least 2 of L, W, and H are equal to 5.

We do not know the value of the third side, we can't get the unique value of V/A. Insufficient.

(2) L, W, and H all have the same value.

Say L = W = H = x

Thus, V/A = LWH / 2(LW + WH + LH) = x^3 / 6x^2 = x/6

Can't get the unique value of V/A. Insufficient.

(1) and (2) together

Since L, W, and H all have the same value and at least 2 of L, W, and H are equal to 5, we have L = W = H = 5 = x.

Thus, V/A = x/6 = 5/6. Sufficient.

The correct answer: C

Hope this helps!

-Jay
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by [email protected] » Tue May 14, 2019 3:12 pm

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Hi All,

We're told that the length, width, and height of a rectangular box, in centimeters, are L, W, and H, respectively, the VOLUME of this box is V cubic centimeters and the TOTAL SURFACE AREA of the 6 sides of this box is A square centimeters. We're asked for the value of V/A. This question is based around some standard Geometry formulas for solids and can be solved by TESTing VALUES.

To start, Volume of a rectangular solid is V = (L)(W)(H) and Total Surface area is SA = 2(L)(W) + 2(L)(H) + 2(W)(H).

(1) At least 2 of L, W, and H are equal to 5.

Fact 1 tells us that 2 (or perhaps all 3) of the dimensions are equal to 5, but that still leads to a number of different answers to the question.
IF...
L = 5, W = 5, H = 5, then the Volume = (5)(5)(5) = 125 and Total Surface Area = (2)(5)(5) + (2)(5)(5) + (2)(5)(5) = 150... so the answer to the question is 125/150 = 5/6.
L = 5, W = 5, H = 1, then the Volume = (5)(5)(1) = 25 and Total Surface Area = (2)(5)(5) + (2)(5)(1) + (2)(5)(1) = 70... so the answer to the question is 25/70 = 5/14.
Fact 1 is INSUFFICIENT

(2) L, W, and H all have the SAME value.

Fact 2 tells us that we're actually dealing with a CUBE, but the answer to the question will still vary depending on the side length.
IF....
L = 5, W = 5, H = 5, then the Volume = (5)(5)(5) = 125 and Total Surface Area = (2)(5)(5) + (2)(5)(5) + (2)(5)(5) = 150... so the answer to the question is 125/150 = 5/6.
L = 1, W = 1, H = 1, then the Volume = (1)(1)(1) = 1 and Total Surface Area = (2)(1)(1) + (2)(1)(1) + (2)(1)(1) = 6... so the answer to the question is 1/6.
Fact 2 is INSUFFICIENT

Combined, we know...
At least 2 of L, W, and H are equal to 5.
L, W, and H all have the SAME value.

When combining the two Facts, it's clear that we're dealing with a cube with a side length of 5, so the answer to the question is 5/6.
Combined, SUFFICIENT

Final Answer: C

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AbeNeedsAnswers wrote:
Thu May 02, 2019 10:44 am
The length, width, and height of a rectangular box, in centimeters, are L, W, and H. If the volume of this box is V cubic centimeters and the total area of the 6 sides of this box is A square centimeters, what is the value of V/A ?

(1) At least 2 of L, W, and H are equal to 5.
(2) L, W, and H all have the same value.

C

Source: Official Guide 2020
Solution:

Question Stem Analysis:


We need to determine the value of V/A. Notice that V = LWH and A = 2(LW + LH + WH).

Statement One Alone:

Without loss of generality, we can let L and W be 5. So, in terms of H, we have V = LWH = 5 x 5 x H = 25H, and A = 2(LW + LH + WH) = 2(5 x 5 + 5H + 5H) = 10(5 + 2H). Therefore, V/A = 25H/[10(5 + 2H)] = 5H/[2(5 + 2H)]. Notice that the value of the ratio changes as the value of H changes. For example, if H = 1, V/A = 5/14. However, if H = 2, V/A = 10/18 = 5/9. Statement one alone is not sufficient.

Statement Two Alone:

We can let L = W = H = s. So, in terms of s, we have V = LWH = s x s x s = s^3, and A = 2(LW + LH + WH) = 2(s x s + s x s + s x s) = 6s^2. Therefore, V/A = s^3 / 6s^2 = s/6. Notice again that the value of the ratio changes as the value of s changes. For example, if s = 1, V/A = 1/6. However, if s = 2, V/A = s/6 = 1/3. Statement two alone is not sufficient.

Statements One and Two Together:

From the two statements, we see that L = W = H = 5. From statement two, we have V/A = s/6. Therefore, s = 5 yields V/A = 5/6. Both statements together are sufficient.

Answer: C

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