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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, 14:25
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Question Stats:

47% (02:34) correct 52% (01:34) wrong based on 114 sessions
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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, 07:07, edited 1 time in total.
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 [#permalink] New post 23 Sep 2004, 20:44
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See attachement, If unclear, let me know.

Regards
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Re: Tunnel hight [#permalink] New post 03 Sep 2010, 08:09
Expert's post
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 3731 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, 21:35
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, 22:40
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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Re: The figure above shows the dimensions of a semicircular [#permalink] New post 25 Oct 2013, 22:41
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Re: The figure above shows the dimensions of a semicircular [#permalink] New post 30 Nov 2013, 13:51
Hi Guys!

I was thinking about extreme cases, such a truck with cylidrical tank. In this case you would not take into account the width of the lane. Hence the maximum height of the truck would be R - 1/2 = 10-1/2 = 9 1/2 (where R is the radius of the semicircular cross).



Unfortuantely, this answer is wrong, but I still think that it is valid. Can somebody explain why I'm wrong? :)
THX!
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File comment: example of the truck I was thinking about
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Re: The figure above shows the dimensions of a semicircular   [#permalink] 30 Nov 2013, 13:51
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