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Senior Manager  Joined: 08 Jul 2004
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The figure above shows the dimensions of a semicircular cross section  [#permalink]

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53 00:00

Difficulty:   75% (hard)

Question Stats: 55% (02:11) correct 45% (02:12) wrong based on 758 sessions

### HideShow timer Statistics 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. 5 ½ ft
B. 7 ½ ft
C. 8 ½ ft
D. 9 ½ ft
E. 10 ft

Attachment: untitled.PNG [ 7.01 KiB | Viewed 45515 times ]

Originally posted by saurya_s on 23 Sep 2004, 15:25.
Last edited by Bunuel on 26 Nov 2018, 04:30, edited 2 times in total.
Renamed the topic and edited the question.
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Nihit wrote: 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

See the diagram attached: 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.

Attachment: untitled.PNG [ 2.53 KiB | Viewed 17488 times ]

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Re: The figure above shows the dimensions of a semicircular cros  [#permalink]

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Nihit wrote:
Attachment:
IMAGE.JPG

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

Say AOB is base of the semi circle.
points X and Y are on edges of the traffic lane (XY=12)
OX=6

Find the height perpendicular to X and touches the semi circle.
= sqrt (6^2+10^2) =8

we need 1/2 ft clearance.

so height must be (8-1/2) =71/2 ft

B.
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See attachement, If unclear, let me know.

Regards
Attachments q4_870.jpg [ 31.85 KiB | Viewed 30631 times ]

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Re: The figure above shows the dimensions of a semicircular cross section  [#permalink]

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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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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 cross section  [#permalink]

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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!
Attachments

File comment: example of the truck I was thinking about fuel_truck_aircraft_3000_gallons_ford_4.jpg [ 22.09 KiB | Viewed 28608 times ]

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MDK wrote:
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!

Hi, late response but it might be helpful to someone else. The reason is because any truck can only have a maximum width = width of the lane, that's what the problem is saying. that means, there are parts of the truck, end points on two sides that can only rise to the height limited by the ceiling of the tunnel at those end points, which is the distance we are trying to find out = 8. hope it's clear.
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Re: The figure above shows the dimensions of a semicircular cross section  [#permalink]

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I have question here, if in the above question they sid that If vehicles must clear the top of the

tunnel by at least 1/2 foot when they are inside the traffic lane. So, they ask about the max

height of the vehicles that allowed to cross the tunnel.We have the radius = 10. So max height of

the vehicle that are allowed to use the tunnel is 10- 0.5 = 9.5 what is the wrong here?
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GMAT 1: 700 Q44 V41 Re: The figure above shows the dimensions of a semicircular cross section  [#permalink]

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23a2012 wrote:
I have question here, if in the above question they sid that If vehicles must clear the top of the

tunnel by at least 1/2 foot when they are inside the traffic lane. So, they ask about the max

height of the vehicles that allowed to cross the tunnel.We have the radius = 10. So max height of

the vehicle that are allowed to use the tunnel is 10- 0.5 = 9.5 what is the wrong here?

The tunnel is rounded and the lane is 12 feet with 4 feet of space on each side, so the vechicle can't be higher than the edges of the lane. Each half of the lane is 6 feet long, so the max height is 1/2 foot smaller than the height of the tunnel 6 feet from the center.

It really helped me to draw it out. As I wrote out the given dimensions, I realized I had a 6-8-10 Pythagorean Triple.
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Re: The figure above shows the dimensions of a semicircular cross section  [#permalink]

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23a2012 wrote:
I have question here, if in the above question they sid that If vehicles must clear the top of the

tunnel by at least 1/2 foot when they are inside the traffic lane. So, they ask about the max

height of the vehicles that allowed to cross the tunnel.We have the radius = 10. So max height of

the vehicle that are allowed to use the tunnel is 10- 0.5 = 9.5 what is the wrong here?

The truck has a width. The entire truck must pass through the semi circular tunnel. I did not get the image I had in mind but look here:

Attachment: 010-Mont-Blanc-Tunnel-Termographic-control-Truck-2011.jpg [ 239.44 KiB | Viewed 26596 times ]

Now imagine that the tunnel is semi circular. If the truck has height of 9.5, its edges will not pass through the sides. Its height needs to be lesser. Since the traffic lane is 12 feet wide and vehicles must stay inside traffic lane, the maximum width of the vehicle will be 12 feet and that will set the limit on the maximum height allowed.
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Re: The figure above shows the dimensions of a semicircular cross section  [#permalink]

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VeritasPrepKarishma wrote:
23a2012 wrote:
I have question here, if in the above question they sid that If vehicles must clear the top of the

tunnel by at least 1/2 foot when they are inside the traffic lane. So, they ask about the max

height of the vehicles that allowed to cross the tunnel.We have the radius = 10. So max height of

the vehicle that are allowed to use the tunnel is 10- 0.5 = 9.5 what is the wrong here?

The truck has a width. The entire truck must pass through the semi circular tunnel. I did not get the image I had in mind but look here:

Attachment:
010-Mont-Blanc-Tunnel-Termographic-control-Truck-2011.jpg

Now imagine that the tunnel is semi circular. If the truck has height of 9.5, its edges will not pass through the sides. Its height needs to be lesser. Since the traffic lane is 12 feet wide and vehicles must stay inside traffic lane, the maximum width of the vehicle will be 12 feet and that will set the limit on the maximum height allowed.

Dear Karishma,thank you +1 KUDOS
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Alt Solution

Attachments 20150529_093316.jpg [ 2.95 MiB | Viewed 26026 times ]

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Even though this question is from GMATprep, I think it's very ambiguous since the question doesn't mention that the car should/could be as wide as the whole lane.
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GMAT 1: 720 Q50 V37 Re: The figure above shows the dimensions of a semicircular cross section  [#permalink]

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maths is easy ...to understand the problem is really hard for a non-native speaker
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How are we certain that the shape of the traffic lane is rectangular? Can't it have a curved head? Re: The figure above shows the dimensions of a semicircular cross section   [#permalink] 23 Dec 2018, 19:10
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