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The figure above shows the dimensions of a semicircular

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The figure above shows the dimensions of a semicircular [#permalink] New post 23 Sep 2004, 15:25
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The figure above shows the dimensions of a semicircular cross section of a one-way tunnel. The single traffic lane is 12 feet wide and is equidistant from the sides of the tunnel. If vehicles must clear the top of the tunnel by at least 1/2 foot when they are inside the traffic lane, what should be the limit on the height of vehicles that are allowed to use the tunnel?

A. 5½ ft
B. 7½ ft
C. 8 ½ ft
D. 9½ ft
E. 10 ft
[Reveal] Spoiler: OA

Last edited by Bunuel on 29 Jan 2012, 08:07, edited 1 time in total.
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 [#permalink] New post 23 Sep 2004, 21:44
See attachement, If unclear, let me know.

Regards
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Re: Tunnel hight [#permalink] New post 03 Sep 2010, 09:09
udaymathapati wrote:
The figure attached shows the dimensions of a semicircular cross section of a one-way tunnel. The single traffic lane is 12 feet wide and is equidistant from the sides of the tunnel. If vehicles must clear the top of the tunnel by at least ½ foot when they are inside the traffic lane, what should be the limit on the height of vehicles that are allowed to use the tunnel?
A. 5½ ft
B. 7½ ft
C. 8 ½ ft
D. 9½ ft
E. 10 ft


See the diagram attached:
Attachment:
untitled.PNG
untitled.PNG [ 5.25 KiB | Viewed 2444 times ]
Rectangle inscribed has the length of traffic lane 12. So max height of vehicle will be 1/2 foot less than the width of this rectangle.

Now, let O be the center of the semi-circle, then OA=radius=20/2=10 and OB=12/2=6 --> AB=\sqrt{OA^2-OB^2}=\sqrt{10^2-6^2}=8.

So max height of the vehicle that are allowed to use the tunnel is 8-0.5=7.5.

Answer: B.
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Re: Tunnel hight [#permalink] New post 03 Sep 2010, 22:35
udaymathapati wrote:
The figure attached shows the dimensions of a semicircular cross section of a one-way tunnel. The single traffic lane is 12 feet wide and is equidistant from the sides of the tunnel. If vehicles must clear the top of the tunnel by at least ½ foot when they are inside the traffic lane, what should be the limit on the height of vehicles that are allowed to use the tunnel?
A. 5½ ft
B. 7½ ft
C. 8 ½ ft
D. 9½ ft
E. 10 ft


My approach:

Equation of the circle -- x^2 + y^2 = a^2 (a is the radius of the circle which is 10 feet.)

The diameter of the semicircle is 20 and the traffic lane is 12 feet wide and located at equal distance from the sides of the tunnel. The width of the traffic lane should be 4 feet away from each of the sides.

If we position the center of the tunnel (center of the semicircle) to overlap exactly on the origin of the x-y coordinate graph then the center of the semicircle would be the origin (0,0) and the end points of the traffic lane would be (-6,0) and (6,0).

Let us take one of the edges of the traffic lane -- (6,0). We need to find distance from the x-axis to the edge of the semi-circle ... that is the y coordinate.

Making use of the equation of the circle -- x^2 + y^2 = 100 .. we already know x coordinate which is 6.

Hence y^2 = 100 - 36.

y^2 = 64. Hence y is 8. Hence the height of the tunnel at the edge of the traffic lane is 8 feet high. Minimum clearance should be 1/2 feet hence the maximum height of the vehicles allowed is 7.5 feet.
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Re: Tunnel hight [#permalink] New post 04 Sep 2010, 23:40
A little correction to my previous post.

The equation of the circle with the given center as (h,k) and radius as r is

(x-h)^2 + (y-k)^2 = r^2.

Since the semicircle is centered at origin (0,0) the equation was noted as x^2 + y^2 = r^2.
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Better solution for Question from GMAT paper test [#permalink] New post 18 May 2012, 04:09
Although the same question has been discussed at tunnel-69768.html. I am posting the same here for better understanding as Bunuel explains the answers in better manner.

The figure above shows the dimensions of a semicircular cross section of a one-way tunnel(dia= 20 ft). The single traffic lane is 12 feet wide and is equidistant from the sides of the tunnel. If vehicles must clear the top of the tunnel by at least ½ foot when they are inside the traffic lane, what should be the limit on the height of vehicles that are allowed to use the tunnel?

A. 5½ ft
B. 7½ ft
C. 8 ½ ft
D. 9½ ft
E. 10 ft
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Re: Better solution for Question from GMAT paper test [#permalink] New post 18 May 2012, 07:30
manjeet1972 wrote:
Although the same question has been discussed at tunnel-69768.html. I am posting the same here for better understanding as Bunuel explains the answers in better manner.

The figure above shows the dimensions of a semicircular cross section of a one-way tunnel(dia= 20 ft). The single traffic lane is 12 feet wide and is equidistant from the sides of the tunnel. If vehicles must clear the top of the tunnel by at least ½ foot when they are inside the traffic lane, what should be the limit on the height of vehicles that are allowed to use the tunnel?

A. 5½ ft
B. 7½ ft
C. 8 ½ ft
D. 9½ ft
E. 10 ft


Merging similar topics. Please ask if anything remains unclear.
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RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. NEW!!!

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set. NEW!!!


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Re: Better solution for Question from GMAT paper test   [#permalink] 18 May 2012, 07:30
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