GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 16 Jun 2019, 21:53

### 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

# Points A and B are 120 km apart. A motorcyclist starts from

Author Message
TAGS:

### Hide Tags

Intern
Joined: 06 Oct 2012
Posts: 5
Re: Points A and B are 120 km apart. A motorcyclist starts from  [#permalink]

### Show Tags

03 Nov 2013, 21:17
the official answer is D, but I am confused
in 3.6 hours the motorcyclist covers (3.6*30) 108 km & the cyclist covers (3.6*10) 36 km, now the hypotenuse is
sqrt(120-108)^2 + sqrt(36)^2 = 48 km

after 4 hours motorcyclist reaches point b and the cyclist covers 40 km, as the road is perpendicular, the distance between two will be 40 km which is less than 48 km at 3.6 hours. so the answer should be E.
Math Expert
Joined: 02 Sep 2009
Posts: 55618
Re: Points A and B are 120 km apart. A motorcyclist starts from  [#permalink]

### Show Tags

03 Nov 2013, 23:17
khosru wrote:
the official answer is D, but I am confused
in 3.6 hours the motorcyclist covers (3.6*30) 108 km & the cyclist covers (3.6*10) 36 km, now the hypotenuse is
sqrt(120-108)^2 + sqrt(36)^2 = 48 km

after 4 hours motorcyclist reaches point b and the cyclist covers 40 km, as the road is perpendicular, the distance between two will be 40 km which is less than 48 km at 3.6 hours. so the answer should be E.

D. $$\sqrt{(120-30*3.6)^2 + (10*3.6)^2} = \sqrt{1,440}$$.
E. $$\sqrt{(120-30*4.6)^2 + (10*4.6)^2} = \sqrt{2,440}$$.
_________________
Director
Joined: 23 Jan 2013
Posts: 547
Schools: Cambridge'16
Re: Points A and B are 120 km apart. A motorcyclist starts from  [#permalink]

### Show Tags

04 Nov 2013, 01:34
1
it is clear that the closer to 4 hours the less the distance. So, it is D
Intern
Joined: 17 Oct 2013
Posts: 39
Location: India
Concentration: Strategy, Statistics
Schools: ISB '17 (A)
GMAT 1: 730 Q49 V40
WE: Analyst (Computer Software)
Re: Points A and B are 120 km apart. A motorcyclist starts from  [#permalink]

### Show Tags

04 Nov 2013, 22:21
I don't think this problems needs calculations of any sort.
Simply visualize a variable right triangle with A toward B forming the base and perpendicularly away from B as the height. The resulting hypotenuse represents the shortest distance between the bikers at any point.
The variable, A towards B (Base) diminishes at a much faster rate than does perpendicularly away from B (The height) because of the significant difference in the speeds of the bikers. Now since the hypotenuse (or the distance between the bikers) is the sum of the squares of the sides, it's least when their combination is minimum ( which obviously happens just before 4 hours or closest to 4 hours).

Guess, visualizing this concept takes much lesser time than reading it (say under a minute!!)
Manager
Joined: 26 Sep 2013
Posts: 190
Concentration: Finance, Economics
GMAT 1: 670 Q39 V41
GMAT 2: 730 Q49 V41
Re: Points A and B are 120 km apart. A motorcyclist starts from  [#permalink]

### Show Tags

22 Nov 2013, 17:19
correct me if I'm wrong, but wouldn't the hypotenuse be at it's shortest when one of the legs is minimized, thus 3.6 would have to be the answer here? The cyclist going north is only going at 10mph, so the leg of the triangle from the cyclist traveling from A to B is getting shorter at a MUCH faster rate than the cyclist traveling north's leg is getting longer. By this logic the longer you wait, the shorter the A-B cyclists leg is going to get, without nearly as much increase in the northward leg. Based on this I said D
Math Expert
Joined: 02 Sep 2009
Posts: 55618
Re: Points A and B are 120 km apart. A motorcyclist starts from  [#permalink]

### Show Tags

23 Nov 2013, 07:09
AccipiterQ wrote:
correct me if I'm wrong, but wouldn't the hypotenuse be at it's shortest when one of the legs is minimized, thus 3.6 would have to be the answer here? The cyclist going north is only going at 10mph, so the leg of the triangle from the cyclist traveling from A to B is getting shorter at a MUCH faster rate than the cyclist traveling north's leg is getting longer. By this logic the longer you wait, the shorter the A-B cyclists leg is going to get, without nearly as much increase in the northward leg. Based on this I said D

No, that's not correct. Why not the answer then 4 hours (E. None)? In this case one of the legs is 0 and the distance is 40 km, which is more than ~38 km in case of 3.6 hours.

Correct solution is here: points-a-and-b-are-120-km-apart-a-motorcyclist-starts-from-92837.html#p717460
_________________
Manager
Joined: 28 Apr 2014
Posts: 206
Re: Points A and B are 120 km apart. A motorcyclist starts from  [#permalink]

### Show Tags

05 May 2014, 10:45
prasannajeet wrote:
Hi Bunuel

Now quadratic function f(x)=ax^2+bx+c reaches its minimum (or maximum when a is negative - not our case), when x=-\frac{b}{2a}=\frac{60*120}{2*1000}=3.6

Rgds
Prasannajeet

I dont think differential / integral calculus must be getting tested in GMAT. If yes , then a single differential is just a proof of point of inflection. The equation would need to be differentiated again to confirm that the point was inflection was maxima or minima
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 9320
Location: Pune, India
Re: Points A and B are 120 km apart. A motorcyclist starts from  [#permalink]

### Show Tags

05 May 2014, 22:11
3
1
himanshujovi wrote:
prasannajeet wrote:
Hi Bunuel

Now quadratic function f(x)=ax^2+bx+c reaches its minimum (or maximum when a is negative - not our case), when x=-\frac{b}{2a}=\frac{60*120}{2*1000}=3.6

Rgds
Prasannajeet

I dont think differential / integral calculus must be getting tested in GMAT. If yes , then a single differential is just a proof of point of inflection. The equation would need to be differentiated again to confirm that the point was inflection was maxima or minima

I would say this question is borderline. GMAT expects you to know how to find the roots of the quadratic using the formula (though it never expects you to actually use it). Also, knowing what the graph of a quadratic looks like is a useful skill on GMAT. We know that when the co-efficient of the x^2 term is positive, it is upward facing so it will have a minimum value. By symmetry, the minimum value will lie right in the center of the roots i.e. it will be the average of the two roots. The two roots are given by
$$\frac{-b + \sqrt{b^2 - 4ac}}{2a} and \frac{-b - \sqrt{b^2 - 4ac}}{2a}$$. Their average is -b/2a.

So you don't actually need to use differential calculus. I wouldn't recommend my students to ignore this question if they were looking at 750.
_________________
Karishma
Veritas Prep GMAT Instructor

Senior Manager
Joined: 08 Apr 2012
Posts: 344
Re: Points A and B are 120 km apart. A motorcyclist starts from  [#permalink]

### Show Tags

12 Sep 2014, 06:06
VeritasPrepKarishma wrote:
himanshujovi wrote:
prasannajeet wrote:
Hi Bunuel

Now quadratic function f(x)=ax^2+bx+c reaches its minimum (or maximum when a is negative - not our case), when x=-\frac{b}{2a}=\frac{60*120}{2*1000}=3.6

Rgds
Prasannajeet

I dont think differential / integral calculus must be getting tested in GMAT. If yes , then a single differential is just a proof of point of inflection. The equation would need to be differentiated again to confirm that the point was inflection was maxima or minima

I would say this question is borderline. GMAT expects you to know how to find the roots of the quadratic using the formula (though it never expects you to actually use it). Also, knowing what the graph of a quadratic looks like is a useful skill on GMAT. We know that when the co-efficient of the x^2 term is positive, it is upward facing so it will have a minimum value. By symmetry, the minimum value will lie right in the center of the roots i.e. it will be the average of the two roots. The two roots are given by
$$\frac{-b + \sqrt{b^2 - 4ac}}{2a} and \frac{-b - \sqrt{b^2 - 4ac}}{2a}$$. Their average is -b/2a.

So you don't actually need to use differential calculus. I wouldn't recommend my students to ignore this question if they were looking at 750.

Hi Karishma
Am I correct in understanding from your explanation that assuming the more time passes the less the distance?
One closes the distance very quickly so the more time passes the less the distance. This is wrong?
If the bicycle would start at the middle do the same straight line, the more time passes the less the distance. Why is this different?
Manager
Joined: 08 Jun 2015
Posts: 102
Re: Points A and B are 120 km apart. A motorcyclist starts from  [#permalink]

### Show Tags

29 Jul 2015, 00:20
jjack0310 wrote:
WholeLottaLove wrote:
Points A and B are 120 km apart. A motorcyclist starts from A to B along straight road AB with speed 30 kmph. At the same time a cyclist starts from B along a road perpendicular to road AB, with a speed of 10 kmph. After how many hours will the distance between them be the least?

See attached graph

This problem is basically looking for the shortest hypotenuse in a triangle comprised of biker on road AB and cyclist moving directly perpendicular to AB from point B. Using the graph, we can plug in numbers into the quadratic formula to see what length C is the shortest.

a^2+b^2=c^2

1 Hour: Biker has traveled 30 km and cyclist has traveled 10 km.
90^2 + 10^2 = c^2
8200 = c^2
We don't need to find the square - we can compare values of c^2

2 Hour: Biker has traveled 60 km and cyclist has traveled 20 km.
60^2+20^ = c^2
4000 = c^2

3 hour: Biker has traveled 90 km and cyclist has traveled 30 km.
30^2 + 30^2 = c^2
1800 = c^2

For each increasing hour, the cyclist and biker move closer together. Out of all the possible answer choices, 3.6 reduces the bikers time the most (and thus the square of a larger speed i.e. 30 km/h) It's also possible to just look at the graph and recognize that the more hours that pass, the shorter the red line between
the biker and cyclist is.

One other thing...technically none of the answer choices are correct. Distance would be minimized when the biker hit B and was 40km away from the cyclist traveling perpendicular from line AB: a^2+b^2=c^2 ===> 0^2+40^2=c^2 ===> 1600 = c^2 so wouldn't E actually be the right answer?

This. I thought the answer would be E, just because the distance will be the least when the motorcyclist reaches point B, because then the distance will be only 40 km.

No, because even after 3.5 hours, the hypotenuse would be 5root58 < 5*8 = 40.
Manager
Joined: 06 Oct 2015
Posts: 91
Re: Points A and B are 120 km apart. A motorcyclist starts from  [#permalink]

### Show Tags

09 Sep 2016, 12:15
wakk0 wrote:
rrsnathan wrote:
gurpreetsingh wrote:
IMO D

Distance traveled by 1st = 30t in t time
distance traveled by 2nd = 10t in t time

now at any instance a right angled triangle is formed whose sides are as follow

1. The distance traveled by 2nd
2. The distance left between the 1st and B
3. Present distance between both the bikers- This is the hypotenuse.

Thus the equation of their distance becomes

d^2 = (10t)^2 + (120-30t)^2

Which is maximum when its differential is 0
=> 20t = 2(120-30t) *3 => t = 3.6

Hi,

Can anyone can explain this part of this solution

""Which is maximum when its differential is 0
=> 20t = 2(120-30t) *3 => t = 3.6 ""

Rrsnathan

gurpreetsingh is drawing on a point from first semester calculus. Let me explain:

In calculus you can take a derivative of a function to calculate the min or max of the function. When you are applying a min or max to a parabola what you are actually finding is the vertex of the parabola (the min or max as the case may be). In order to find this you take the first derivative of the function, set it equal to zero, and then solve for your variable. This solution will produce the answer (the min or max of the parabola) for the problem. If you would like an explanation of this process let me know and I will post it. However, if you are a bit rusty on calculus Bunuel's answer is probably the better course to take.

Hi,
I am not good at calculus. But I know a little about differentiation. Would you, please, tell me in respect of what you are differentiating?
Manager
Joined: 17 May 2015
Posts: 248
Re: Points A and B are 120 km apart. A motorcyclist starts from  [#permalink]

### Show Tags

23 Oct 2016, 03:52
NaeemHasan wrote:
Hi,
I am not good at calculus. But I know a little about differentiation. Would you, please, tell me in respect of what you are differentiating?

We are differentiating W.R.T. $$t$$. You can solve this question without Calculus. Please check previous posts.
Intern
Joined: 20 Mar 2016
Posts: 31
GMAT 1: 530 Q33 V28
Re: Points A and B are 120 km apart. A motorcyclist starts from  [#permalink]

### Show Tags

02 May 2017, 21:42
Shouldn't the answer be E? For example at 3.7 hours the distance would be lesser than what it was in 3.6 hrs.

Thanks,
Jat
_________________
Manager
Joined: 27 Jul 2016
Posts: 77
GMAT 1: 730 Q49 V40
WE: Consulting (Consulting)
Points A and B are 120 km apart. A motorcyclist starts from  [#permalink]

### Show Tags

23 May 2018, 01:08
Hi,
To be honest, when I saw the question for the first time, I had no idea to solve it. ( I roughly remember that calculus might work, thus I tried it. )
Here is my method to solve the question
T - in hour
D - distance between cars

I started with forming a distance equation
(120-30T)^2 + (10T)^2 = D^2

Then, I expanded terms

120^2 - 2*30T*120 + 900T^2 + 100T^2 = D^2

I knew that, in this case, I did not have to care about D value, so I just applied differentiation:
*right side is zero because it's the vertex(might be max or mix value). In this case, I was sure that it gives time at minimum distance.
-2*30*120 +2000T=0

T=3.6 hrs

Best regards,
G.
Senior Manager
Joined: 09 Jun 2014
Posts: 269
Location: India
Concentration: General Management, Operations
Schools: Tuck '19
Re: Points A and B are 120 km apart. A motorcyclist starts from  [#permalink]

### Show Tags

20 Oct 2018, 21:45
einstein10 wrote:
Points A and B are 120 km apart. A motorcyclist starts from A to B along straight road AB with speed 30 kmph. At the same time a cyclist starts from B along a road perpendicular to road AB, with a speed of 10 kmph. After how many hours will the distance between them be the least?

A. 3 hours
B. 3.4 hours
C. 3.5 hours
D. 3.6 hours
E. None

Spoiler: :: OA
D, please provide workable method that can solve this in 2mins, i solved but it takes more than 3mins. can any one help.
thanks

Distance traveled by 1st = 30t in t time
distance traveled by 2nd = 10t in t time

now at any instance a right angled triangle is formed whose sides are as follow

1. The distance traveled by 2nd
2. The distance left between the 1st and B
3. Present distance between both the bikers- This is the hypotenuse.

Thus the equation of their distance becomes

d^2 = (10t)^2 + (120-30t)^2

Which is maximum when its differential is 0
=> 20t = 2(120-30t) *3 => t = 3.6

Can someone please explain how the last equation has come from first.I was trying to do first differential but I am no getting the same result.
Senior Manager
Joined: 09 Jun 2014
Posts: 269
Location: India
Concentration: General Management, Operations
Schools: Tuck '19
Re: Points A and B are 120 km apart. A motorcyclist starts from  [#permalink]

### Show Tags

20 Oct 2018, 21:46
prabsahi wrote:
einstein10 wrote:
Points A and B are 120 km apart. A motorcyclist starts from A to B along straight road AB with speed 30 kmph. At the same time a cyclist starts from B along a road perpendicular to road AB, with a speed of 10 kmph. After how many hours will the distance between them be the least?

A. 3 hours
B. 3.4 hours
C. 3.5 hours
D. 3.6 hours
E. None

Spoiler: :: OA
D, please provide workable method that can solve this in 2mins, i solved but it takes more than 3mins. can any one help.
thanks

Distance traveled by 1st = 30t in t time
distance traveled by 2nd = 10t in t time

now at any instance a right angled triangle is formed whose sides are as follow

1. The distance traveled by 2nd
2. The distance left between the 1st and B
3. Present distance between both the bikers- This is the hypotenuse.

Thus the equation of their distance becomes

d^2 = (10t)^2 + (120-30t)^2

Which is maximum when its differential is 0
=> 20t = 2(120-30t) *3 => t = 3.6

Can someone please explain how the last equation has come from first.I was trying to do first differential but I am no getting the same result.

Manager
Joined: 17 May 2015
Posts: 248
Re: Points A and B are 120 km apart. A motorcyclist starts from  [#permalink]

### Show Tags

20 Oct 2018, 23:25
1
Hi prabsahi,

Quote:
Thus the equation of their distance becomes

d^2 = (10t)^2 + (120-30t)^2

Which is maximum when its differential is 0
=> 20t = 2(120-30t) *3 => t = 3.6

After differentiating with 't' and equating to zero, we get following:

$$2*10t*10 + 2*(120-30t)*(-30) \Rightarrow 200t + 1800t - 120*60 = 0 \Rightarrow 2000t = 120*60 \Rightarrow t = 3.6$$

Alternatively, you can first expand the expression and then differentiate. i.e.

$$d^2 = 100t^2 + 900t^2 - 2*30t*120 + 120^2 = 1000t^2 - 2*30t*120 + 120^2$$

If we differentiate above expression w.r.t. 't' and equate it equal to zero, we get

$$2000t - 2*30*120 =0 => t = 3.6$$

In general, it is better to avoid calculus for GMAT question. You can use the following results. These results can be obtained using the method of completing the square.

Minimum or Maximum Value of a Quadratic Equation

The minimum and maximum value of a quadratic equation $$f(x) = ax^2 + bx + c ; (a \neq 0)$$ occurs at

$$x = \frac{-b}{2a}$$

• if $$a > 0$$, then the minimum value is $$f(\frac{-b}{2a})$$
• if $$a < 0$$, then the maximum value is $$f(\frac{-b}{2a})$$

Hope this helps.
Senior Manager
Joined: 09 Jun 2014
Posts: 269
Location: India
Concentration: General Management, Operations
Schools: Tuck '19
Re: Points A and B are 120 km apart. A motorcyclist starts from  [#permalink]

### Show Tags

20 Oct 2018, 23:31
ganand wrote:
Hi prabsahi,

Quote:
Thus the equation of their distance becomes

d^2 = (10t)^2 + (120-30t)^2

Which is maximum when its differential is 0
=> 20t = 2(120-30t) *3 => t = 3.6

After differentiating with 't' and equating to zero, we get following:

$$2*10t*10 + 2*(120-30t)*(-30) \Rightarrow 200t + 1800t - 120*60 = 0 \Rightarrow 2000t = 120*60 \Rightarrow t = 3.6$$

Alternatively, you can first expand the expression and then differentiate. i.e.

$$d^2 = 100t^2 + 900t^2 - 2*30t*120 + 120^2 = 1000t^2 - 2*30t*120 + 120^2$$

If we differentiate above expression w.r.t. 't' and equate it equal to zero, we get

$$2000t - 2*30*120 =0 => t = 3.6$$

In general, it is better to avoid calculus for GMAT question. You can use the following results. These results can be obtained using the method of completing the square.

Minimum or Maximum Value of a Quadratic Equation

The minimum and maximum value of a quadratic equation $$f(x) = ax^2 + bx + c ; (a \neq 0)$$ occurs at

$$x = \frac{-b}{2a}$$

• if $$a > 0$$, then the minimum value is $$f(\frac{-b}{2a})$$
• if $$a < 0$$, then the maximum value is $$f(\frac{-b}{2a})$$

Hope this helps.

Indeed this is very helpful .Thanks a lot!!
Re: Points A and B are 120 km apart. A motorcyclist starts from   [#permalink] 20 Oct 2018, 23:31

Go to page   Previous    1   2   [ 38 posts ]

Display posts from previous: Sort by