The square root of integer x is equal to the sum of the cubes of y and

Question

The square root of integer x is equal to the sum of the cubes of y and z. If the squares of y and z are each less than 10, what is the greatest possible value of x?

A. 2916
B. 3981
C. 3982
D. 3999
E. 4000

chetan2u · Accepted Answer

Hi...
Let the values be √10 each.  here the values will be just LESS than 10, so our ACTUAL answer will be slightly less than the answer we get here  
So sum of cubes is  2*10^{\frac{3}{2}} 
Square of this is  4*10^3=4000 ..

But the actual ans will be slightly LESS than 4000 and an integer..
So 3999
D

Blackbox · Answer

Chetan - Thanks for responding. Although, I wish you hadn't used "yellow" to highlight the text against an almost yellow background. Nevertheless, here is my approach and I don't know if I arrived at the wrong answer. Given, \sqrt{x} = y^3 + z^3 And is also given, y^2 < 10 z^2 < 10 So, the biggest value that y and z could each take is 3. Since, the question asks for the highest value, I have assumed the highest value for y and z (i.e, 3). \sqrt{x} = 3^3 + 3^3 => \sqrt{x} = 27 + 27 => \sqrt{x} = 54 and squaring both sides, we get: x = {54}^2 x = 2916. Answer = A Please correct if I am wrong.

JeffTargetTestPrep · Answer

We can create the following equation:

√x = y^3 + z^3

Since the squares of y and z are each less than 10, y < √10 and z < √10, which implies y^3 < 10√10 and z^3 < 10√10.

Thus:

√x < 10√10 +10√10

√x < 20√10

x < 4000

Since x is an integer, the greatest value of x is 3999.

Answer: D

Blackbox · Answer

Beats me why I should "root" the y instead of assuming a possible highest value for y in  y^2  e.g., y= 3 ?

shouri · Answer

Y and z need not be integers

Sent from my SM-G610F using  GMAT Club Forum mobile app

chetan2u · Answer

Hi,

Sorry for the colour. Didn't realise it.

You are WRONG when you read it as an integer.
It is nowhere given as an integer.
So if √10=3.16227766.., the value could be 3.16227765..
Hence we find answer by taking it as 10 and finding one value lesser INTEGER.

Hope it helps

GTExl · Answer

perfect squares less than 10 are 1 ,4,9 for which value of y,z could be 1 ,2 ,3

but for greatest possible value of x we will take y= z= 3

now √x = y^3+z^3 =3^3 + 3^3

√x = 27+27 = 54

x= 54^2
x=2916

Answer A

FB2017 · Answer

Perfect squares less than 10 are 1 ,4,9 for which value of y,z could be 1 ,2 ,3
but for greatest possible value of x we will take y= z= 3
now √x = y^3+z^3 =3^3 + 3^3
√x = 27+27 = 54
x= 54^2
x=2916
Option A.

Problem states square root of integer x is equal to the sum of the cubes of y and z, hence to achieve that we cannot have y & z to be non integer values as we frame the equation x=y^3+z^3=10√10+10√10, as √10 when simplified would give a decimal component as well. Or in other words y & z cannot be √10 each.
Please advise.

GTExl · Answer

Hi  FB2017

Although I did the same as you did, but I was wrong. below is explanation-
 
we are not saying x=10√10+10√10 rather it is x<10√10+10√10 i.e x<20√10 (Note: if we take x=20√10 then we will get x=4000)

as x<20√10 we are saying that x<4000 i.e x=3999 as x should be an integer.

Kudo please if this helps you.

shashankism · Answer

\sqrt{x} = y^3 + z^3 
 y^2 <10 , z^2 <10 
 -> y <\sqrt{10}, z <\sqrt{10}

So,  \sqrt{x} < \sqrt{10}^3 + \sqrt{10}^3 
 \sqrt{x} < 20*\sqrt{10} 
 x < (20*\sqrt{10})^2 
 x<4000 
 So x = 3999

Answer D

sahilvijay · Answer

Ans is clearly D

consider max values
y^2=10 means y^6 =1000 and same for z^6=1000. or y=z
x^0.5 = y^3+z^3
square both sides
x= y^6+z^6+2(yz)^3
 = 1000+1000+2(z^2)^3
 = 1000=1000+2(z^6)
 =1000+1000+2(1000)
 = 4000
but y and z are less than 10
therefore x is less than 4000 and integer value less than 4000 is 3999.

Give kudos take kudos

Fdambro294 · Answer

(1st)

Sum of Cubes:

(A)^3 + (B)^3 = (A + B) (A^2 - AB + B^2)

(2nd) if we are given that the square root of X equals the SUM of Y cubed and Z cubed, since we can not take the square root of a negative value, both Y and Z must be positive values

(3rd)

Given that

(Y)^2 < 10

(Z)^2 < 10

We can take the square root of both sides of the inequality without worrying about the negative value because Z and Y must be positive

0 < Y < sqrt(10)

0 < Z < sqrt(10)

(Lastly)

Assume that Y and Z are both equal to the value of their boundary ——-> sqrt(10)

Plugging in sqrt(10) into the Sum of Cubes formula given above we get

sqrt(X) = (2 * sqrt(10)) * (10 - sqrt(100) + 10)

And

sqrt(X) = (2 * sqrt(10)) * (20 - 10)

And

sqrt(X) = 20 * sqrt(10)

——squaring both sides to eliminate the square root we get——

X = (400) * (10)

Or

X = 4000

However, the values of Y and Z were LESS than the square root of 10, which we plugged into the Sum of Cubes formula.

Thus, the value for X must be less than this amount

X < 4000

D is the largest answer below

Posted from my mobile device

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

The square root of integer x is equal to the sum of the cubes of y and

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

The square root of integer x is equal to the sum of the cubes of y and

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards