Last visit was: 27 Apr 2026, 18:06 It is currently 27 Apr 2026, 18:06
Home
Messages
Tests
Schools
Events
Close
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
Your Progress

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
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Go to My Error Log Learn more
Close
Request Expert Reply
Confirm Cancel
Back to Forum
Create Topic
805+ (Hard)|   Geometry|      
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 27 Apr 2026
Posts: 109,929
Own Kudos:
811,619
 [3]
Given Kudos: 105,914
Products:
GMAT Club Tests Forum Quiz
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 109,929
Kudos: 811,619
 [3]
Alice and Bob live 10 miles apart. One day Alice looks due north from 09 May 2019, 21:32
Kudos
Add Kudos
3
Bookmarks
Bookmark this Post
00:00
Start Timer
Pause Timer
Resume Timer
Show Answer
Hide
Show
History
E
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

95% (hard)

Question Stats:

31% (03:20) correct 69% (02:56) wrong based on 59 sessions

History

Date
Time
Result
Not Attempted Yet
Alice and Bob live 10 miles apart. One day Alice looks due north from her house and sees an airplane. At the same time Bob looks due west from his house and sees the same airplane. The angle of elevation of the airplane is 30° from Alice's position and 60° from Bob's position. Which of the following is closest to the airplane's altitude, in miles?

(A) 3.5

(B) 4

(C) 4.5

(D) 5

(E) 5.5
ShowHide Answer
Official Answer
E
User avatar
nick1816
User avatar
Retired Moderator
Joined: 19 Oct 2018
Last visit: 12 Mar 2026
Posts: 1,841
Own Kudos:
8,513
Given Kudos: 707
Location: India
Posts: 1,841
Kudos: 8,513
Alice and Bob live 10 miles apart. One day Alice looks due north from 14 May 2019, 03:42
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Let airplane's altitude = x

Let Bob house is at B
tan60=x/BC
\(BC=x/\sqrt{3}\)

Let Alice house is at A
tan30=x/AC
\(AC=x*\sqrt{3}\)

AC^2 + BC^2 = AB^2
3x^2+x^2/3 =10^2
10x^2/3=100
x^2=30
x= 5.48
Attachments

dfgdhd.png
dfgdhd.png [ 5.53 KiB | Viewed 3653 times ]

User avatar
Mugdho
User avatar
Joined: 27 Feb 2019
Last visit: 11 Nov 2023
Posts: 93
Own Kudos:
250
Given Kudos: 495
Posts: 93
Kudos: 250
Alice and Bob live 10 miles apart. One day Alice looks due north from 03 Feb 2021, 10:55
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Bunuel
Alice and Bob live 10 miles apart. One day Alice looks due north from her house and sees an airplane. At the same time Bob looks due west from his house and sees the same airplane. The angle of elevation of the airplane is 30° from Alice's position and 60° from Bob's position. Which of the following is closest to the airplane's altitude, in miles?

(A) 3.5

(B) 4

(C) 4.5

(D) 5

(E) 5.5
Show more


Bunuel Please give a solution of this problem.

Posted from my mobile device
User avatar
Mugdho
User avatar
Joined: 27 Feb 2019
Last visit: 11 Nov 2023
Posts: 93
Own Kudos:
250
Given Kudos: 495
Posts: 93
Kudos: 250
Alice and Bob live 10 miles apart. One day Alice looks due north from 03 Feb 2021, 13:24
Kudos
Add Kudos
Bookmarks
Bookmark this Post
chetan2u
Please provide a valid solution to this problem.
Posted from my mobile device
User avatar
chetan2u
User avatar
GMAT Expert
Joined: 02 Aug 2009
Last visit: 27 Apr 2026
Posts: 11,229
Own Kudos:
45,031
 [3]
Given Kudos: 335
Status:Math and DI Expert
Location: India
Concentration: Human Resources, General Management
GMAT Focus 1: 735 Q90 V89 DI81
Products:
My Reviews
Expert
Expert reply
GMAT Focus 1: 735 Q90 V89 DI81
Posts: 11,229
Kudos: 45,031
 [3]
Alice and Bob live 10 miles apart. One day Alice looks due north from 04 Feb 2021, 03:45
2
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
Bunuel
Alice and Bob live 10 miles apart. One day Alice looks due north from her house and sees an airplane. At the same time Bob looks due west from his house and sees the same airplane. The angle of elevation of the airplane is 30° from Alice's position and 60° from Bob's position. Which of the following is closest to the airplane's altitude, in miles?

(A) 3.5

(B) 4

(C) 4.5

(D) 5

(E) 5.5
Show more

Mugdho

Let O be the place where airplane is there. The point directly under it is point C, which gives altitude as OC=x.

Now we have 3 right angled triangles ABC, where A and B are positions of Alice and Bob. Also ABC is created on the ground.
\(\triangle AOC\) is 30-60-90 triangle as A looks up 30 degree to see the plane. So OC:AC:OA=1:\(\sqrt{3}\):2=x:\(\sqrt{3}\)x:2x
\(\triangle OBC\) is 30-60-90 triangle as B looks up 60 degree to see the plane. So BC:OC:OB=1:\(\sqrt{3}\):2=\(\frac{x}{\sqrt{3}}\):x:\(2*\frac{x}{\sqrt{3}}\)

Finally, \(\triangle ABC\) is also a right angled triangle with sides BC and AC as \(\frac{x}{\sqrt{3}}\) and \({x*\sqrt{3}}\)
So the hypotenuse \(AB^2=10^2=100=(\frac{x}{\sqrt{3}})^2+({x*\sqrt{3}})^2=\frac{x^2}{3}+3x^2=\frac{10x^2}{3}\)
\(300=10x^2........x^2=30..........25<30<36.......5^2<x^2<6^2.........5<x<6\)

E
Attachments

11.png
11.png [ 18.81 KiB | Viewed 2893 times ]

User avatar
Mugdho
User avatar
Joined: 27 Feb 2019
Last visit: 11 Nov 2023
Posts: 93
Own Kudos:
250
Given Kudos: 495
Posts: 93
Kudos: 250
Alice and Bob live 10 miles apart. One day Alice looks due north from 05 Feb 2021, 06:11
Kudos
Add Kudos
Bookmarks
Bookmark this Post
chetan2u
Bunuel
Alice and Bob live 10 miles apart. One day Alice looks due north from her house and sees an airplane. At the same time Bob looks due west from his house and sees the same airplane. The angle of elevation of the airplane is 30° from Alice's position and 60° from Bob's position. Which of the following is closest to the airplane's altitude, in miles?

(A) 3.5

(B) 4

(C) 4.5

(D) 5

(E) 5.5

Mugdho

Let O be the place where airplane is there. The point directly under it is point C, which gives altitude as OC=x.

Now we have 3 right angled triangles ABC, where A and B are positions of Alice and Bob. Also ABC is created on the ground.
\(\triangle AOC\) is 30-60-90 triangle as A looks up 30 degree to see the plane. So OC:AC:OA=1:\(\sqrt{3}\):2=x:\(\sqrt{3}\)x:2x
\(\triangle OBC\) is 30-60-90 triangle as B looks up 60 degree to see the plane. So BC:OC:OB=1:\(\sqrt{3}\):2=\(\frac{x}{\sqrt{3}}\):x:\(2*\frac{x}{\sqrt{3}}\)

Finally, \(\triangle ABC\) is also a right angled triangle with sides BC and AC as \(\frac{x}{\sqrt{3}}\) and \({x*\sqrt{3}}\)
So the hypotenuse \(AB^2=10^2=100=(\frac{x}{\sqrt{3}})^2+({x*\sqrt{3}})^2=\frac{x^2}{3}+3x^2=\frac{10x^2}{3}\)
\(300=10x^2........x^2=30..........25<30<36.......5^2<x^2<6^2.........5<x<6\)

E
Show more

Hey chetan2u Thank you so much. In the picture I understand Alice is looking "due north" but how Bob is looking "due west"... rather looks like Bob is looking "North-West"..I Don't understand here

Posted from my mobile device
User avatar
chetan2u
User avatar
GMAT Expert
Joined: 02 Aug 2009
Last visit: 27 Apr 2026
Posts: 11,229
Own Kudos:
45,031
 [2]
Given Kudos: 335
Status:Math and DI Expert
Location: India
Concentration: Human Resources, General Management
GMAT Focus 1: 735 Q90 V89 DI81
Products:
My Reviews
Expert
Expert reply
GMAT Focus 1: 735 Q90 V89 DI81
Posts: 11,229
Kudos: 45,031
 [2]
Alice and Bob live 10 miles apart. One day Alice looks due north from 05 Feb 2021, 06:35
2
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Mugdho
chetan2u
Bunuel
Alice and Bob live 10 miles apart. One day Alice looks due north from her house and sees an airplane. At the same time Bob looks due west from his house and sees the same airplane. The angle of elevation of the airplane is 30° from Alice's position and 60° from Bob's position. Which of the following is closest to the airplane's altitude, in miles?

(A) 3.5

(B) 4

(C) 4.5

(D) 5

(E) 5.5

Mugdho

Let O be the place where airplane is there. The point directly under it is point C, which gives altitude as OC=x.

Now we have 3 right angled triangles ABC, where A and B are positions of Alice and Bob. Also ABC is created on the ground.
\(\triangle AOC\) is 30-60-90 triangle as A looks up 30 degree to see the plane. So OC:AC:OA=1:\(\sqrt{3}\):2=x:\(\sqrt{3}\)x:2x
\(\triangle OBC\) is 30-60-90 triangle as B looks up 60 degree to see the plane. So BC:OC:OB=1:\(\sqrt{3}\):2=\(\frac{x}{\sqrt{3}}\):x:\(2*\frac{x}{\sqrt{3}}\)

Finally, \(\triangle ABC\) is also a right angled triangle with sides BC and AC as \(\frac{x}{\sqrt{3}}\) and \({x*\sqrt{3}}\)
So the hypotenuse \(AB^2=10^2=100=(\frac{x}{\sqrt{3}})^2+({x*\sqrt{3}})^2=\frac{x^2}{3}+3x^2=\frac{10x^2}{3}\)
\(300=10x^2........x^2=30..........25<30<36.......5^2<x^2<6^2.........5<x<6\)

E

Hey chetan2u Thank you so much. In the picture I understand Alice is looking "due north" but how Bob is looking "due west"... rather looks like Bob is looking "North-West"..I Don't understand here

Posted from my mobile device
Show more

The triangle ABC is a right angled triangle on the ground. So if A to C is due north, B to C will be due west as C is 90 degree.
Now O is right above C, so whatever direction C is to B, O will also be in the same direction.

You are wrongly visualising O to north of C. But OC is like a pole with O as top.
Moderators:
Bunuel
Math Expert
109929 posts
Krunaal
Tuck School Moderator
852 posts