In cross section, a tunnel that carries one lane of one-way traffic is

Video solution from Quant Reasoning:

Agreed. At first I thought a truck 2.4 m wide at 4 m height wont be able to pass the tunnel that is 4.2 m high. But quick use of pythagoras theorem shows that at 4 m height a truck 2.56 m wide can pass.

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Bunuel, Can u please explain how the tunnel would look like.That is where I am not able to comprehend the question.

Hi suhasreddy ,

Just draw a semicircle and consider the base of the semi circle as the road.
So for any truck to pass through it, we would require both it's width and height.
Hence C is the ans.

There you go bud!

i THINK the answer should be E.
Reason:
Option 2. says MAXIMUM height of the truck is 4.
If we assume the truck is conical in shape and having only one point of 4 Meter height then yes it can pass through a tunnel of radius 4.2 M
WHat if the truck is rectangular in shape?.Then it will have a 2 point of height 4 M separated by distance of 2.4 M (its width )
.In that case its both edges will not pass through the semicircular tunnel of max heigh 4.2 M

Makes sense guys?

Why is it not necessary to perform a calculation?

I would think you would need to know the height of the tunnel, 1.2m away from the center of the tunnel (semi circle).

(1.2)^2 + 4^2 is less than (4.2)^2

Bunuel: Can you please suggest similar problems

Somewhat similar questions:
https://gmatclub.com/forum/in-the-figur ... 88102.html
https://gmatclub.com/forum/the-figure-a ... 00337.html
https://gmatclub.com/forum/a-right-circ ... 35548.html
https://gmatclub.com/forum/in-a-hemisph ... 95387.html

Good point made buddy! EVen in the case of rectangular truck with 4m height & 2.4m width, the truck can pass through the tunnel. Lets consider the truck is in the middle of the semicircle tunnel. we can form a right angle triangle with half of the width 2.4/2=1.2 m as base and 4m as height. The hypotenuse will be approximately 4.17, which is less than 4.2m. So the truck can pass through easily.

Answer is C

Because Data Sufficiency is awesome!

If this was a Problem Solving question, you'd be 100% correct. But because this is DS, it actually doesn't matter whether the truck can or can't fit through the tunnel. It just matters whether we have enough info to figure it out, if we wanted to. If we knew all of the dimensions of the truck and the tunnel, we could figure out whether it would fit just by doing some math, so having that information is sufficient.

This is a tough thing to get your head around, because you're used to solving math problems where the question is 'can it fit?'. But in DS, that's not really the question you're being asked. You're being asked "have we provided enough information that a smart person would be able to determine whether it could fit?". If someone could figure it out, it's sufficient. You don't have to do the actual 'figuring out' part.

On the contrary, IT IS necessary to perform the calculations to know the answer to that question.

Using Pythagore, the height of the tunnel 1.2m (4.2m/2) from its center can be found to be 4.05m. Because, the truck's maximum height is 4m (< 4.05m), the tunnel will be able to accomodate the truck. However, if while looking for the height of the tunnel 1.2m from its center we found ANY number lower than 4m -- say 3.90m --, because we are given only maximum height and not actual height, it would have been impossible to know if the truck would have been able to go through the tunnel or not.

In that particular case of height = 3.9m, if the truck is 3.95m high, it will be able to go through the tunnel. If it is only 3.85m high, it will not be able to go through the tunnel. The answer would have been E

Given the length of calculations to arrive at the tunnel's height at 1.2m form the center, I think the stated difficulty level of this problem should be readjusted. Bunuel

I agree with Aardwolf. It is absolutely necessary to calculate in order to solve this Q since the statements in question refer to maximum height and width. So a given rectangle is the maximum possible area of the truck, but the truck could be any other shape to that more easily fits within the tunnel. If the Q specifically said the truck is in the shape of a rectangle, you could avoid calc. but this is not the case either.

Preferred Approach:

Taking both statements together, the maximum possible area of the truck is a rectangle with height of 4m and width of 2.4m. Since it's a one-lane road, the best possible way to fit the truck is if it drives perfectly midway of the tunnel such that that bottom-centre of the truck is equidistant from the walls.

We then need to split this rectangle into two halves (each with width=2.4/2=1.2, height=4) and identify if the diagonal of that rectangle (referred to below as x) is greater than the radius (r) of the tunnel.



Since it is a right-angled triangle with base of 1.2 and height of 4, x is calculated by pythag: \(x=\sqrt{1.2^2+4^2}\approx{4.18}\). As per the Question, \(r=4.2\).

The tunnel will always accommodate the truck, regardless of the truck's actual shape if \(r>x\). So the question effectively becomes is \(4.2>\sqrt{1.2^2+4^2}\approx{4.18}\)? The answer to this Q is YES hence C is correct.

HOWEVER if the answer to this Q is NO then, it does not mean the truck will not fit (see above diagram). It merely means the truck MAY not fit, depending on the shape of the truck. There are shapes other than a rectangle that would could have a \(maximum\) width of 2.4m and height of 4m. Hence there would be insufficient information to answer the question and the answer would be E in these circumstances.

Would really appreciate if someone could shed light on how to more efficiently approximate that 4.2 is indeed greater than \(\sqrt{1.2^2+4^2}\). This part of the question caused me the most difficulty given time constraints. Bunuel

If you have \sqrt{2} memorized than you know it is equal to 1.4. Using this information I estimated up and made 1.2 -->1.4.

We originally have \sqrt{(1.2)^2+(4)^2} --estimating up for 1.2--> \sqrt{(1.4)^2+(4)^2} =\sqrt{2+16/square_root]

=[square_root]18} = 3\sqrt{2} = (3)*(1.4) = 4.2 ---> since I estimated up. The actual number must be smaller than 4.2

It would be good to have \sqrt{2} & \sqrt{3} memorized.

I agree with ngmat12 about the fact that we should do the necessary calculations to evaluate the two statements together.



Instead of comparing \(\sqrt{1.2^2+4^2}\) to 4.2, we can compare the squares of these two positive terms.

\(\left(\sqrt{1.2^2+4^2}\right)^2=1.2^2+4^2=1.44+16=17.44\)

\(4.2^2=17.64\)

\(17.44<17.64 \implies \sqrt{1.2^2+4^2}<4.2\)

I totally diagree with difficulty level of this question and the way in which many people are solving it JeffTargetTestPrep AbdurRakib . If you really want to be sure C is correct, you must check to see if the answer would be true if the truck was a maximized rectangle/cuboid. Thus you would have to do pythagroean and theorem to find hypotenus
ie SQRT((W/2)^2+H^2) or with values SQRT(1.2^2+4^2)

Then you would have to make sure hypotenuse is less than the radius of 4.2 to be sure the truck could always pass through. IMO this would make problem more difficult.

Bunuel, would you agree?

Those disagreeing with Jeff, Bunuel and the Mgmat guy about the method and difficulty levels are not asking a question.
If I were the mathists I wouldn't be able to respond bcos you aren't asking any questions. You're labouring yourselves.

Since you think the top edge of the truck might scrape the roof of the circular tunnel or impede the truck and then you realize that another math could help you figure out if that will happen or not using the infos in 1 and 2, then u don't need to do that math. So the math doesn't matter. No matter how further u try to complicate the question, you still have the information in both to determine if u can answer yes or no.

The only difficult aspect of this question is figuring out the imagery in the question stem asap.
It's a "real life problem" question, so u need to see yourself as the truck driver to do that fast.


I suggest you ask the experts some questions when things are not clear. It's not just a name. They are really experts. If you doubt, just ask them the questions anyway and see for yourself.

Hope it helps.

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