Join us for MBA Spotlight – The Top 20 MBA Fair      Schedule of Events | Register

 It is currently 04 Jun 2020, 14:02

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# If x^2 + y^2 = 100, and x≥0, and y≥0, the maximum value of x

Author Message
TAGS:

### Hide Tags

Manager
Joined: 09 Feb 2013
Posts: 111
If x^2 + y^2 = 100, and x≥0, and y≥0, the maximum value of x  [#permalink]

### Show Tags

Updated on: 11 Feb 2013, 13:30
7
45
00:00

Difficulty:

65% (hard)

Question Stats:

54% (01:42) correct 46% (01:47) wrong based on 620 sessions

### HideShow timer Statistics

If x^2 + y^2 = 100, and x≥0, and y≥0, the maximum value of x + y is

A. Less than 10
B. Greater than or equal to 10 and less than 14
C. Greater than 14 and less than 19
D. Greater than 19 and less than 23
E. Greater than 23

Originally posted by emmak on 11 Feb 2013, 09:56.
Last edited by Bunuel on 11 Feb 2013, 13:30, edited 1 time in total.
Renamed the topic and edited the question.
Math Expert
Joined: 02 Sep 2009
Posts: 64275
Re: If x^2 + y^2 = 100, and x≥0, and y≥0, the maximum value of x  [#permalink]

### Show Tags

11 Feb 2013, 13:44
11
23
emmak wrote:
If x^2 + y^2 = 100, and x≥0, and y≥0, the maximum value of x + y is

A. Less than 10
B. Greater than or equal to 10 and less than 14
C. Greater than 14 and less than 19
D. Greater than 19 and less than 23
E. Greater than 23

$$x^2 + y^2 = 100$$ --> $$(x+y)^2-2xy=100$$ --> $$(x+y)^2=100+2xy$$ --> $$x+y=\sqrt{100+2xy}$$.

We need to maximize $$x+y$$, thus we need to maximize $$\sqrt{100+2xy}$$. To maximize $$\sqrt{100+2xy}$$ we need to maximize $$xy$$.

Now, for given sum of two numbers, their product is maximized when they are equal, hence from $$x^2 + y^2 = 100$$ we'll have that $$x^2y^2$$ (or which is the same xy) is maximized when $$x^2=y^2$$.

In this case $$x^2 + x^2 = 100$$ --> $$x=\sqrt{50}$$.

So, we have that $$\sqrt{100+2xy}$$ is maximized when $$x=y=\sqrt{50}$$, so the maximum value of $$\sqrt{100+2xy}$$ is $$\sqrt{100+2*\sqrt{50}*\sqrt{50}}=\sqrt{200}\approx{14.1}$$.

_________________
Manager
Joined: 08 Dec 2012
Posts: 60
Location: United Kingdom
WE: Engineering (Consulting)
Re: If x^2 + y^2 = 100, and x≥0, and y≥0, the maximum value of x  [#permalink]

### Show Tags

11 Feb 2013, 14:05
11
1
I tried a more crude method:

$$x^2 + y^2 = 100$$

We know $$6^2 + 8^2$$ is one of the options which will lead to x+y = 14. So options A & B are out.

We also know $$7^2 + 7^2$$ we get just $$98<100$$. So something slightly more than $$7$$ i.e. $$7.1$$ or whatever that will lead to the answer of $$100$$. So the maximum value of $$x+y$$ is just over $$14$$. Any other combination of x & y cannot be more than this value of just over 14. So answer is C.
##### General Discussion
Manager
Joined: 02 Jan 2013
Posts: 54
GMAT 1: 750 Q51 V40
GPA: 3.2
WE: Consulting (Consulting)
Re: If x^2 + y^2 = 100, and x≥0, and y≥0, the maximum value of x  [#permalink]

### Show Tags

13 Feb 2013, 08:31
7
An elegant solution is looking at this problem geometrically.

x^2 + y^2 = 100 is a circumference with radius 10 and center in (0,0).

The equation x + y = k (where we want to maximize k) is a set of infinite parallel negative slope (-45 deg) lines.

Now note that k is the Y INTERSECT of all such lines. In order to maximize this y intersect (and therefore maximize k) we need to find the line of the set that is tangent to the circunference in the 1st quadrant.

Now this becomes a (fairly easy) geometry problem.

Drawing it out you'll have no problem seeing that
Y intersect = k = RADIUS / Sin(45 deg) = 10.sqrt(2) = approx 14,1

Posted from my mobile device
Intern
Joined: 23 Feb 2013
Posts: 8
Re: If x^2 + y^2 = 100, and x≥0, and y≥0, the maximum value of x  [#permalink]

### Show Tags

11 Mar 2013, 17:17
Dear Bunuel,

Can you please explain this statement,"for given sum of two numbers, their product is maximized when they are equal". Can you give some theory and examples for this statement to be true ? I would be happy to understand this. Thank you.
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 10497
Location: Pune, India
Re: If x^2 + y^2 = 100, and x≥0, and y≥0, the maximum value of x  [#permalink]

### Show Tags

11 Mar 2013, 19:43
12
6
emmak wrote:
If x^2 + y^2 = 100, and x≥0, and y≥0, the maximum value of x + y is

A. Less than 10
B. Greater than or equal to 10 and less than 14
C. Greater than 14 and less than 19
D. Greater than 19 and less than 23
E. Greater than 23

You can use the logical approach to get the answer within seconds.

First you need to understand how squares work - As you go to higher numbers, the squares rise exponentially (obviously since they are squares!)
What I mean is
2^2 = 4
3^2 = 9
4^2 = 16
(as we increase the number by 1, the square increases by more than the previous increase. From 2^2 to 3^2, the increase is 9-4= 5 but from 3^2 to 4^2, the increase is 16 - 9 = 7... As we go to higher numbers, the squares will keep increasing more and more.)

If we want to keep the square at 100 but maximize the sum of the numbers, we should try and make the numbers as small as possible so that their contribution in the square doesn't make the other number very small i.e. if you take one number almost 10, the other number will become very very small and the sum will not be maximized. If one number is made small, the other will become large, hence both numbers should be equal to maximize the sum.
So the square of each number should 50 i.e. each number should be a little more than 7.
Both numbers together will give us something more than 14.

_________________
Karishma
Veritas Prep GMAT Instructor

Math Expert
Joined: 02 Sep 2009
Posts: 64275
Re: If x^2 + y^2 = 100, and x≥0, and y≥0, the maximum value of x  [#permalink]

### Show Tags

12 Mar 2013, 05:50
2
2
Backbencher wrote:
Dear Bunuel,

Can you please explain this statement,"for given sum of two numbers, their product is maximized when they are equal". Can you give some theory and examples for this statement to be true ? I would be happy to understand this. Thank you.

If a+c=k, then the product ab is maximized when a=b.

For example, if given that a+b=10, then ab is maximized when a=b=5 --> ab=25.

Hope it's clear.
_________________
Retired Moderator
Joined: 18 Sep 2014
Posts: 1068
Location: India
Re: If x^2 + y^2 = 100, and x≥0, and y≥0, the maximum value of x  [#permalink]

### Show Tags

04 Jun 2015, 05:50
VeritasPrepKarishma wrote:
emmak wrote:
If x^2 + y^2 = 100, and x≥0, and y≥0, the maximum value of x + y is

A. Less than 10
B. Greater than or equal to 10 and less than 14
C. Greater than 14 and less than 19
D. Greater than 19 and less than 23
E. Greater than 23

You can use the logical approach to get the answer within seconds.

First you need to understand how squares work - As you go to higher numbers, the squares rise exponentially (obviously since they are squares!)
What I mean is
2^2 = 4
3^2 = 9
4^2 = 16
(as we increase the number by 1, the square increases by more than the previous increase. From 2^2 to 3^2, the increase is 9-4= 5 but from 3^2 to 4^2, the increase is 16 - 9 = 7... As we go to higher numbers, the squares will keep increasing more and more.)

If we want to keep the square at 100 but maximize the sum of the numbers, we should try and make the numbers as small as possible so that their contribution in the square doesn't make the other number very small i.e. if you take one number almost 10, the other number will become very very small and the sum will not be maximized. If one number is made small, the other will become large, hence both numbers should be equal to maximize the sum.
So the square of each number should 50 i.e. each number should be a little more than 7.
Both numbers together will give us something more than 14.

Sorry to open this post after so long time again but i got a bit confused reg the solution.
I agree with the solution that maximizing sum is by making both the numbers x and y equal resulting in value greater than 14.
but what about checking whether the value is less than 19 or not.
GMAT Club Legend
Status: GMATINSIGHT Tutor
Joined: 08 Jul 2010
Posts: 4049
Location: India
GMAT: QUANT EXPERT
Schools: IIM (A)
GMAT 1: 750 Q51 V41
WE: Education (Education)
Re: If x^2 + y^2 = 100, and x≥0, and y≥0, the maximum value of x  [#permalink]

### Show Tags

04 Jun 2015, 06:03
Mechmeera wrote:
VeritasPrepKarishma wrote:
emmak wrote:
If x^2 + y^2 = 100, and x≥0, and y≥0, the maximum value of x + y is

A. Less than 10
B. Greater than or equal to 10 and less than 14
C. Greater than 14 and less than 19
D. Greater than 19 and less than 23
E. Greater than 23

You can use the logical approach to get the answer within seconds.

First you need to understand how squares work - As you go to higher numbers, the squares rise exponentially (obviously since they are squares!)
What I mean is
2^2 = 4
3^2 = 9
4^2 = 16
(as we increase the number by 1, the square increases by more than the previous increase. From 2^2 to 3^2, the increase is 9-4= 5 but from 3^2 to 4^2, the increase is 16 - 9 = 7... As we go to higher numbers, the squares will keep increasing more and more.)

If we want to keep the square at 100 but maximize the sum of the numbers, we should try and make the numbers as small as possible so that their contribution in the square doesn't make the other number very small i.e. if you take one number almost 10, the other number will become very very small and the sum will not be maximized. If one number is made small, the other will become large, hence both numbers should be equal to maximize the sum.
So the square of each number should 50 i.e. each number should be a little more than 7.
Both numbers together will give us something more than 14.

Sorry to open this post after so long time again but i got a bit confused reg the solution.
I agree with the solution that maximizing sum is by making both the numbers x and y equal resulting in value greater than 14.
but what about checking whether the value is less than 19 or not.

Hi Mechmeera,

The maximum value has been calculated when x = y = $$5\sqrt{2}$$

i.e. x+y will have maximum value of $$5\sqrt{2}$$ + $$5\sqrt{2}$$ = $$10\sqrt{2}$$ = 14.14

Since the value of x+y can't exceed beyond the calculated value therefore

we can conclude that it is less that 19 and greater than 14.
_________________
Prosper!!!
GMATinsight .............(Bhoopendra Singh and Dr.Sushma Jha)
e-mail: info@GMATinsight.com l Call : +91-9999687183 / 9891333772
One-on-One Skype classes l Classroom Coaching l On-demand Quant course l Admissions Consulting
Check website for most affordable Quant on-Demand course 2000+ Qns (with Video explanations)
Our SUCCESS STORIES: From 620 to 760 l Q-42 to Q-49 in 40 days l 590 to 710 + Wharton l
ACCESS FREE GMAT TESTS HERE:22 FREE (FULL LENGTH) GMAT CATs LINK COLLECTION
Math Expert
Joined: 02 Aug 2009
Posts: 8623
Re: Problem Solving: Number Properties  [#permalink]

### Show Tags

31 Dec 2017, 19:43
BetaRayBryan wrote:
If $$x^2 + y^2 = 100$$, $$x\geq{0}$$ and $$y\geq{0}$$, the maximum value of x + y must be which of the following?

A) Less than 10

B) Greater than or equal to 10 and less than 14

C) Greater than 14 and less than 19

D) Greater than 19 and less than 23

E) Greater than 23

Would someone be able to walk me through this? I'm sort of unclear on the reasoning behind the correct answer.. Thank you so much!

Hi

$$x^2+y^2=100=10^2$$..
Least- when the number are far away so 10 and 0, x+y=10
Max- when X and y are close by.. 7^2=49 so both as 7 gives us 7^2+7^2=98
And x+y =7+7=14, so ans will be slightly MORE than 14
C
_________________
Intern
Joined: 25 Dec 2015
Posts: 4
Re: If x^2 + y^2 = 100, and x≥0, and y≥0, the maximum value of x  [#permalink]

### Show Tags

07 Feb 2018, 09:59
Thank you all, I got the same answer but with bit of confused, I AGREED with answer 14.1 but the answer also given as > 14 to <19. So, it implies that the answer could be anywhere from 14.1 to 18.9? I understand that 14.1 is the possible answer, then why we have answer given <19?

Thank you.
Intern
Joined: 07 Sep 2016
Posts: 11
Location: India
Concentration: Technology, Finance
GMAT 1: 700 Q49 V36
If x^2 + y^2 = 100, and x≥0, and y≥0, the maximum value of x  [#permalink]

### Show Tags

09 Apr 2018, 08:49
caioguima wrote:
An elegant solution is looking at this problem geometrically.

x^2 + y^2 = 100 is a circumference with radius 10 and center in (0,0).

The equation x + y = k (where we want to maximize k) is a set of infinite parallel negative slope (-45 deg) lines.

Now note that k is the Y INTERSECT of all such lines. In order to maximize this y intersect (and therefore maximize k) we need to find the line of the set that is tangent to the circunference in the 1st quadrant.

Now this becomes a (fairly easy) geometry problem.

Drawing it out you'll have no problem seeing that
Y intersect = k = RADIUS / Sin(45 deg) = 10.sqrt(2) = approx 14,1

Posted from my mobile device

chetan2u
Bunuel
Could you explain this solution in further detail please. Thanks.
GMAT Club Legend
Joined: 18 Aug 2017
Posts: 6314
Location: India
Concentration: Sustainability, Marketing
GPA: 4
WE: Marketing (Energy and Utilities)
Re: If x^2 + y^2 = 100, and x≥0, and y≥0, the maximum value of x  [#permalink]

### Show Tags

22 Feb 2019, 09:02
emmak wrote:
If x^2 + y^2 = 100, and x≥0, and y≥0, the maximum value of x + y is

A. Less than 10
B. Greater than or equal to 10 and less than 14
C. Greater than 14 and less than 19
D. Greater than 19 and less than 23
E. Greater than 23

we can use the formula of a^2=b^2+c^2
knowing 10^2 = 6^2 + 8^2
so max value would be 8+6 ; 14
option C
Non-Human User
Joined: 09 Sep 2013
Posts: 15071
Re: If x^2 + y^2 = 100, and x≥0, and y≥0, the maximum value of x  [#permalink]

### Show Tags

14 Mar 2020, 16:35
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Re: If x^2 + y^2 = 100, and x≥0, and y≥0, the maximum value of x   [#permalink] 14 Mar 2020, 16:35