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In cross section, a tunnel that carries one lane

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jjjinapinch Senior | Next Rank: 100 Posts Default Avatar
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In cross section, a tunnel that carries one lane

Post Mon Jul 31, 2017 6:38 am
In cross section, a tunnel that carries one lane of one-way traffic is a semicircle with radius 4.2 m. Is the tunnel large enough to accommodate the truck that is approaching the entrance to the tunnel?
(1) The maximum width of the truck is 2.4 m
(2) The maximum height of the truck is 4 m

Official Guide question
Answer: C

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Post Wed Aug 09, 2017 11:19 am
jjjinapinch wrote:
In cross section, a tunnel that carries one lane of one-way traffic is a semicircle with radius 4.2 m. Is the tunnel large enough to accommodate the truck that is approaching the entrance to the tunnel?
(1) The maximum width of the truck is 2.4 m
(2) The maximum height of the truck is 4 m

Official Guide question
Answer: C
We are given that a tunnel in the shape of a semicircle has a radius of 4.2 meters. Since the tunnel has a radius of 4.2 meters and hence a diameter of 8.4 meters, its maximum height is 4.2 meters and its maximum width is 8.4 meters. We need to determine whether the tunnel is large enough to fit a particular truck.

Statement One Alone:

The maximum width of the truck is 2.4 m.

Although we know the maximum width of the truck, without knowing the maximum height of the truck, we cannot determine whether it can fit through the tunnel. Statement one alone is not sufficient to answer the question.

Statement Two Alone:

The maximum height of the truck is 4 m.

Although we know the maximum height of the truck, without knowing the maximum width of the truck, we cannot determine whether it can fit through the tunnel. Statement two alone is not sufficient to answer the question. We can eliminate answer choice B.

Statements One and Two Together:

From the given information and the statements, we know that the width of the tunnel is 8.4 m, the maximum height of the tunnel is 4.2 m, the maximum width of the truck is 2.4 m, and the maximum height of the truck is 4 m. Since we have all of the necessary dimensions of the truck and the tunnel, we have the information to determine whether the truck would fit through the tunnel. Note that it is not necessary to perform the calculations, but the answer is yes, the truck can pass through the tunnel.

Answer: C

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Post Mon Jul 31, 2017 11:15 pm
jjjinapinch wrote:
In cross section, a tunnel that carries one lane of one-way traffic is a semicircle with radius 4.2 m. Is the tunnel large enough to accommodate the truck that is approaching the entrance to the tunnel?

(1) The maximum width of the truck is 2.4 m
(2) The maximum height of the truck is 4 m

Official Guide question
Answer: C
We are given that the entrance of the shape of the tunnel is a semi-circle that has a radius of 4.2 meters.

We need to ensure whether the tunnel is large enough to accommodate the truck.

To answer this question we need to ensure that the truck's cross section that is of rectangular shape can be accommodated if the truck follows the middle-most path inside the tunnel.

Since the radius of the tunnel is 4.2 m, the diameter of the tunnel = 2*4.2 = 8.4 m. For the truck to pass through the tunnel, the width of the truck < 8.4 m.

Since the radius of the tunnel is 4.2 m, the height of the tunnel at the center = 4.2 m (Maximum). As we move from the center of the semi-circular cross section of the tunnel to the wall of the tunnel, the height of the tunnel keeps decreasing. For the truck to pass through the tunnel, the maximum height of the truck < the height of the tunnel at the vertical edge of the truck.

Statement 1: The maximum width of the truck is 2.4 m.

Clearly insufficient as we do not know the height of the truck. If the height of the truck is too less, it will pass; however, if it is too high, it will not. Insufficient.

Statement 2: The maximum height of the truck is 4 m.

Clearly insufficient as we do not know the width of the truck. If the width of the truck is too less, it will pass; however, if it is too high, it will not. Insufficient.

Statement 1 & 2 combined:

Since the maximum width of the truck is 2.4 m, which is less than the diameter (8.4 m) of the tunnel, it is not a problem.

We must now calculate the height of the tunnel where the vertical edge of the truck is positioned; say that position is V.

Assume that the truck follows the middle-most path.

Height of the tunnel at position V = √[(Radius)^2 - (half of width of the truck)^2]

= √[(4.2)^2 - (1.2)^2]

Case 1: If √[(4.2)^2 - (1.2)^2] < 4 (Max. height of the truck); the answer is YES.
Case 2: If √[(4.2)^2 - (1.2)^2] = 4 (Max. height of the truck); the answer is YES.
Case 3: If √[(4.2)^2 - (1.2)^2] > 4 (Max. height of the truck); the answer is NO.

We need not calculate the value of √[(4.2)^2 - (1.2)^2]; whatever be the value of √[(4.2)^2 - (1.2)^2], we are sure that either the TRUCK would PASS or the TRUCK would NOT PASS. The answer is unique. Sufficient.

The correct answer: C

For your curiosity to know whether the truck would pass through or not (Not need though in a DS question), let's calculate the value of √[(4.2)^2 - (1.2)^2 = √[(4.2 + 1.2)*(4.2 - 1.2)] = √[5.4 * 3] = √16.2

Since √16.2 < 4 (Max. height of the truck), the truck would pass.

Hope this helps!

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