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

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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?

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 [ 5.25 KiB | Viewed 11966 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.

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

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25 Oct 2013, 23:41

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

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30 Nov 2013, 14:51

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!

Attachments

File comment: example of the truck I was thinking about

fuel_truck_aircraft_3000_gallons_ford_4.jpg [ 22.09 KiB | Viewed 5099 times ]

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

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19 Nov 2014, 17:33

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.

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

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11 Feb 2015, 12:57

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.

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

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11 Feb 2015, 22:24

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Expert's post

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 3091 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. _________________

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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