The dimensions of a rectangular solid are 4 inches, 5 inches

I am pretty sure about the thing marked in red. It should state rectangular solid and not the sphere........if that is the case find the answer below.....

cube with side greater than 4 cannot fit in the cuboid (4,5 & 8)

so cube has volume of 4^3 = 64

volume of cuboid = 4*5*8 = 160

area not occupied by cube = 96

so the answer is 64:96 or 2:3 (its perfect 2:3) not approx as stated in your OA

hope this helps !!!

The cube must have a side of 4 inches to fit in rectangular solid. Now as the cube is inscribed in sphere then the radius of this sphere equals to half of the cube's diagonal --> \(d=\sqrt{4^2+4^2+4^2}=4\sqrt{3}\), so radius of the sphere is \(r=\frac{d}{2}=2\sqrt{3}\)

Volume of the cube will be \({volume}_{cube}={side}^3=4^3=64\);
Volume of the sphere will be \({volume}_{sphere}=\frac{4}{3}\pi{r^3}=\frac{4}{3}\pi{24\sqrt{3}}={32\sqrt{3}\pi}\).

The ratio of the volume of the cube to the volume within the sphere that is not occupied by the cube is \(\frac{64}{32\sqrt{3}\pi-64}\), and after some calculations you'll get that this ratio is closer to 10/17 than to any other ratios in answer choices.

Answer: A.

P.S. I really doubt that GMAT would offer such a problem, as it requires lengthy calculations and bad approximations (\(\pi\), \(\sqrt{3}\)).

thanx nusma for posting sucha question ...however pls explain why do we have
third 4^2 .... in calculation of diagonal .. I am surely missing something

BTW what is source of such difficult problem

I m not able to understand, why the side of the cube is 4 ? It is not mentioned that the cube is inside the rectangular box

Yes, the question should mention that the cube can be entirely placed in the rectangular solid.

If cube is to be fitted in sphere, it will take 10 minutes to fit it within it

if cube is to be fitted in Rectangular solid, only then its possible to solve it within 2 mins.

Any case it is 700 plus level question.

My answer in scene 1 is - 10/17
My answer in scene2 is - 2/3

Hi Bunnel,

Why will the answer be different if I use "a" as the side of the cube.

a^3 gets cancelled, and i get, 1/(4*pi*3^(1/2)) = 7/(88*3^(1/2)-7) = .04

Please, show your work so that it's clear how you get the expression in your post.

Also, please use proper formatting: rules-for-posting-please-read-this-before-posting-133935.html#p1096628

Hi Bunuel,
Y the cube must have a side of 4 inches only to fit in? Please help me understand.

Because the dimensions of a rectangular solid in which we should place a cube are 4 inches, 5 inches, and 8 inches. If the cube's side is more than 4 it won't fit in the rectangular solid.

I was stuck. The answer here is 0.49 (on calculation). 2/5 = 0.4 and 10/17=0.59. Not sure which one is closer? : /

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