Hi Brent, thank you for the excellent explanation. BrentGMATPrepNow

One question: if we keep the dimensions of the top layer (6 x 4) the same and gradually make the can larger, the level would drop. Would this change the volume?
For instance, if we look at a can with a diameter of 100, the amount of gas would be very low within that can, but the top layer would still be 6 x 4 with a depth of 2.

It is a DS question so some imagination can give us the answer without solving at all. This is what the cylinder looks like:

The question stem tells us that Height = 2 and Length = 6.

Stmnt 1: Diameter = 4

This means the tank is half full. We can find the total volume of the tank using the formula of volume of a cylinder and divide it by 2. We will get the volume of gasoline. Sufficient alone.

Stmnt 2: The area of the rectangle is 24.
We know that the Length is 6 so the other side of the rectangle (as visible in the pic) would be 4. Now imagine this: I have a horizontal line of length 4. If I draw a line from its centre of length 2, I will get a defined arc (which we see is a semi circle but that is irrelevant anyway since it is a DS question).


If there is only one way in which we can make the arc, it means it has defined dimensions and hence the area of this segment can be calculated. Hence the volume of gasoline will be area of this segment * Length of the cylinder.
Sufficient alone.

Answer (D)

Hi VeritasKarishma

I am confused about the orientation of the figure. Shouldn't "circular ends oriented vertically" tell us that the cylinder would be a verticalle right cylinder? I reached the correct answer by assuming that? Am I missing anything?

A student asked me to explain why statement 2 is sufficient. So,......

Target question: What is the volume of gasoline in the tank?

Statement 2: The top surface of the gasoline forms a rectangle that has an area of 24 square feet.
We know that the tank is 6 ft long
Let x = the width of the surface
We have the following:



Area of rectangle = (base)(height)
So, we can write: 24 = (6)(x)
Solve get: x = 4
So, the diagram looks like this:


IMPORTANT: For geometry Data Sufficiency questions, we’re typically checking to see whether the statements "lock" a particular angle, length, or shape into having just one possible measurement. This concept is discussed in much greater detail in the following video: https://www.gmatprepnow.com/module/gmat ... /video/884

So we want to determine whether there is exactly one tank size such that when the gasoline is 2 feet deep, the surface of the gasoline is 4 feet wide.

So let's start with some random tank like this...

... and fill it with gasoline to a depth of 2 feet

When we measure the width of the surface, we get 3 feet

So, this particular tank is too small.

Let's try slightly bigger tank. Here's one.

Still too small.

As we can see, if we gradually make the tank bigger and bigger and bigger, there will be EXACTLY ONE tank such that the width of the surface is 4 feet.


Since there is EXACTLY ONE tank that meets the given conditions, we know that statement 2 is SUFFICIENT

Answer: D

ASIDE: Some people might be wondering, "Sure, but what is the actual volume of the gasoline in the tank?"
Fortunately, we don't need to find the actual volume. We need to only demonstrate that we COULD find the volume.
One way to do this would be to use geometric properties and formulas. But we can also find the volume in other ways.
For example, if I had very precise measuring equipment and a way to construct cylindrical tanks of any shape, then I COULD keep testing actual tanks until I find one that meets the given conditions. Then, I'd fill that tank with gasoline to a depth of 2 feet. Then I'd pour that gasoline into a very precise measuring cup. Done!!

Cheers,
Brent

Every time we change the diameter of the tank, the width of water (when the tank is 2 in deep) changes.
For example, if we take a tank with diameter 100 and fill it to a depth of 2 feet, the width of the water will be 28 ft (not 4 feet)

BrentGMATPrepNow - how did you calculate 28 feet in yellow ?

IMPORTANT: For geometry Data Sufficiency questions, we’re typically checking to see whether the statements "lock" a particular angle, length, or shape into having just one possible measurement. This concept is discussed in much greater detail in the following video: https://www.gmatprepnow.com/module/gmat ... /video/884

So we want to determine whether there is exactly one tank size such that when the gasoline is 2 feet deep, the surface of the gasoline is 4 feet wide.

So let's start with some random tank like this...

... and fill it with gasoline to a depth of 2 feet

When we measure the width of the surface, we get 3 feet

So, this particular tank is too small.

Let's try slightly bigger tank. Here's one.

Still too small.

As we can see, if we gradually make the tank bigger and bigger and bigger, there will be EXACTLY ONE tank such that the width of the surface is 4 feet.


Since there is EXACTLY ONE tank that meets the given conditions, we know that statement 2 is SUFFICIENT

Answer: D

ASIDE: Some people might be wondering, "Sure, but what is the actual volume of the gasoline in the tank?"
Fortunately, we don't need to find the actual volume. We need to only demonstrate that we COULD find the volume.
One way to do this would be to use geometric properties and formulas. But we can also find the volume in other ways.
For example, if I had very precise measuring equipment and a way to construct cylindrical tanks of any shape, then I COULD keep testing actual tanks until I find one that meets the given conditions. Then, I'd fill that tank with gasoline to a depth of 2 feet. Then I'd pour that gasoline into a very precise measuring cup. Done!!

Cheers,
Brent

Hi BrentGMATPrepNow - Just wanted to confirm three quick points based on the above 'lock' mentioned in the yellow specifically

(i) if i understand, what you are saying is -- given the length of the cylinder is constant (=6 feet). Really the only change one can make is on the diameter of the cylinder.

The larger the diameter, for a given height of 2 feet of gasoline -- the depth of the gasoline is constantly going to change

So, a cylinder of 6 feet length, gasoline height of 2 feet -- if you want a depth of 4 feet.... there is only one diameter of a cylinder that will work (as one can see, the longer the diameter of the cylinder, for a given height of 2 feet ) --> that is 'lock' you are mentioning above

(ii) This 'lock' was only true as long as the cylinders length of 6 feet is constant, correct ?

What if the cylinders length of 6 feet WAS NOT constant ?

I dont think this 'lock' strategy would have worked.

Just confirming

(iii) How are you so sure from your 4 circles above -- the 'lock' was when the diameter of the cylinder was 4 feet specifically ?

KarishmaB while solving this question, I realized the following. Please correct me if I am wrong:

Diameter = 2 x radius so from (2) we get the depth as "2" and the width has "4" and in a circle, the diameter is the ONLY chord whose half-length can be drawn? In other words, the diameter is the ONLY chord from which we can draw a line from its center touching the circumference of the circle.


In other words, if I know that "2" and "4" are the measurements given I cannot have a chord other than the diameter that can be twice of "2"?

Also, I really liked what you and a few other experts did for (2). You guys just "fixed" the semi-circle that is formed from "4" and "2" thus realizing that the area of this semicircle can be found. Then we multiply it with length 6 and arrive at our answer