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Math Expert V
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The four vertices of a face of a cube have coordinates (a, b), (a, d),  [#permalink]

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The four vertices of a face of a cube have coordinates (a, b), (a, d), (c, b), and (c, d). If a = 2 and c is an even number between 6 and 11, which of the following could be the surface area of the cube?

A. 96
B. 150
C. 294
D. 384
E. 600

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Re: The four vertices of a face of a cube have coordinates (a, b), (a, d),  [#permalink]

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Let the C point to be 8, then , the cube has a side of 6.

Surface area= 6a^2=6*6*6= 216. Answer not there.

If the C point be 10, then side=8. surface area= 6*8*8=384

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Re: The four vertices of a face of a cube have coordinates (a, b), (a, d),  [#permalink]

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Bunuel wrote:
The four vertices of a face of a cube have coordinates (a, b), (a, d), (c, b), and (c, d). If a = 2 and c is an even number between 6 and 11, which of the following could be the surface area of the cube?

A. 96
B. 150
C. 294
D. 384
E. 600

TSA of cuboid = 6s^2
so from given info
c can be either 8 , 10 ; so value of cube side = 6,8
from the given options we can determine value of side
A. 96=s = 4
B. 150; s= 5
C. 294;s=7
D. 384;s=8
E. 600;s=10

option D is correct ; and value of c = 10
IMO D
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Re: The four vertices of a face of a cube have coordinates (a, b), (a, d),  [#permalink]

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can anybody please draw the cube on a coordinate plane ?
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GMAT 1: 780 Q51 V46 Re: The four vertices of a face of a cube have coordinates (a, b), (a, d),  [#permalink]

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Hello,

Greetings for the day!

This question is a fairly straightforward question on Mensuration. This tests your knowledge of basic mensuration concepts like the properties of a cube and the formula to calculate the Surface area of a cube. You can also utilise your knowledge of divisibility rules to try and eliminate some options, although in this particular question, this technique may not be particularly useful.

A cube is a three dimensional figure in which all the surfaces are squares. Hence, the length, breadth and height are all equal in a cube. A cube is made up of 6 square surfaces. Therefore,
Surface Area of a Cube = 6a^2 , where a represents the side of the cube.
Now, let us look at the question at hand. We know that the four vertices given represent the vertices of the face of the given cube. Hence, they should represent the four corners of a square which makes up a face of the given cube.
(a,b) , (a, d), (c, b) and (c,d) are the four vertices given. Of these, (a, b) and (a,d) will lie on the same vertical line since they have their x co-ordinates same. Similarly, (c,b) and (c,d) will lie on the same vertical line which is different from the previous line, since a≠c.
Using a similar analogy, we can say that (a,b) and (c,b) will be on the same horizontal line since they have their y co-ordinates same, while, (a,d) and (c,d) will also lie on the same horizontal line, albeit different from the previous one.
We also know that a=2. Hence, (a,b) and (a,d) can be substituted by (2,b) and (2,d), while plotting the problem on the co-ordinate plane. It is said that c is an even number between 6 and 11. Hence, the two possible values of c are 8 and 10. Substituting these, we have four possibilities for the other two vertices given:
(c,b) = (8,b)
(c,d) = (8,d)
(c,b) = (10,b)
(c,d) = (10,d)

Using the data given above, plotting a diagram on the co-ordinate plane will give us a figure which looks like the one below:

Attachment: The four vertices of a face of a cube have coordinates (a, b), (a, d),.jpg [ 32.98 KiB | Viewed 440 times ]

From the diagram above, we can conclude that the side of the cube can be either 6 units or 8 units.
If the side of the cube is 6 units, a = 6 and Surface area = 6 * 6^2 = 216 sq.units.
If the side of the cube is 8 units, a = 8 and Surface area = 6 * 8^2 = 384 sq.units.
Remember that the question is asking us to find out a possible value of the surface area. Hence, 384 could be the surface area of the cube.
Hence, the answer option is D.
In questions like these, since the surface area of a cube has to be a multiple of 6, answer options which are not multiples of 6 can be eliminated. However, in this question, since all options were multiples of 6, we did not follow this approach. Also, since the question is a could be question, it is not necessary to find a unique answer. It is enough to calculate all possible answers which satisfy the conditions and match them to the answer options to zero in on the final answer.

Hope this helps!
Cheers,
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Re: The four vertices of a face of a cube have coordinates (a, b), (a, d),  [#permalink]

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Bunuel wrote:
The four vertices of a face of a cube have coordinates (a, b), (a, d), (c, b), and (c, d). If a = 2 and c is an even number between 6 and 11, which of the following could be the surface area of the cube?

A. 96
B. 150
C. 294
D. 384
E. 600

We see that c can be either 8 or 10. Let’s analyze both numbers.

If c = 8, then an edge of the cube would have length of 8 - 2 = 6, and the surface area of the cube would then be 6(6)^2 = 216. However, 216 is not one of the choices.

If c = 10, then an edge of the cube would have length of 10 - 2 = 8, and the surface area of the cube would then be 6(8)^2 = 384. We see that 384 is one of the choices.

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