A right circular cone, twice as tall as it is wide at its

Question

A right circular cone, twice as tall as it is wide at its greatest width, is pointing straight down. The cone is partially filled with water, which is dripping out of a tiny hole in the cone's tip at a rate of 2 cubic centimeters per hour. If the water were to continue to drip out at this rate, how much longer would it take for the cone to empty, assuming that no water is added to the cone and that there is no loss of water from the cone by any other means?

(1) The top surface of the water in the cone is currently \(9\pi\) square centimeters in area.
(2) The top surface of the water in the cone currently is exactly 4 centimeters below the cone's top, measuring vertically.

Show Answer

Official Answer

A

(1) The top surface of the water in the cone is currently  9\pi  square centimeters in area.
(2) The top surface of the water in the cone currently is exactly 4 centimeters below the cone's top, measuring vertically.

Bunuel · Accepted Answer

Disagree. A right circular cone, twice as tall as it is wide at its greatest width, is pointing straight down. The cone is partially filled with water, which is dripping out of a tiny hole in the cone's tip at a rate of 2 cubic centimeters per hour. If the water were to continue to drip out at this rate, how much longer would it take for the cone to empty, assuming that no water is added to the cone and that there is no loss of water from the cone by any other means? https://gmatclub.com/forum/download/file.php?id=21632 H - height of cone, R radius of cone. h - height of water in cone, r radius of top surface of water in cone. (1) The top surface of the water in the cone is currently 9\pi square centimeters in area. Top surface area of water = 9\pi=\pi*r^2 --> r=3 --> \frac{R}{H}=\frac{r}{h}=\frac{1}{4} --> h=12 --> V=\frac{1}{3}*\pi*r^2*h=36*\pi cubic centimeters. Leak rate 2 cubic centimeters per hour --> time=\frac{36\pi}{2} hours. Sufficient. (2) The top surface of the water in the cone currently is exactly 4 centimeters below the cone's top, measuring vertically. Not sufficient we know that H=h+4, but h can be any value and thus the Volume can be any. Answer: A. Untitled.png

jax91 · Answer

(1) The top surface of the water in the cone is currently 9pi square centimeters in area. -- sufficient on its own (we can get D)
(2) The top surface of the water in the cone currently is exactly 4 centimeters below the cone's top, measuring vertically. -- sufficient on its own ( we have H)

As the ratio of diameter:hieght of any sub-cone formed in this cone will be 1:2

So we need either D or H to get the volume.

IMO D.

svikram · Answer

It cant be A cause the question says, e cone is partially filled with water.

A can be true only when the cone is fully filled.

testprepDublin · Answer

If we know the ratio for height:width for any volume in a cone, then that ratio applies to all volumes. This rule is due to the fact that the angles in the cone stay constant when the volume changes. Statement 1 gives us the area at the top of the water. This allows us to find the water volume, using the height:width ratio provided, and subsequently the rate of leaking. A is the answer.

kartik222 · Answer

hi Bunuel,
how did you get to the ratio:
R/H=r/h=1/4 ??
Can you please elaborate?
thanks!!

maaadhu · Answer

kartik,

similar triangle property...

R/H = r/h

since 2R=H

r/h = 1/2

so h = 6.

Paris75 · Answer

Hi,

have a small question: it is stated that " The cone is partially filled with water, which is dripping out of a tiny hole in the cone's tip at a rate of 2 cubic centimeters per hour "

It has been said that : the leak rate 2 cubic centimeters per hour --> 36pi/2. Meaning that it will take 18pi hours to fill the cone.

But, the cone could be half full. or 3/4 full. We don't know!

Therefore, the result will be different since they ask : " how much longer would it take for the cone to empty , assuming that no water is added to the cone and that there is no loss of water from the cone by any other means"

It could be 18pi/2 or 18pi/4!

Where did i miss something? Plz explain!

Thanks!

Bunuel · Answer

36\pi  is the volume of water in the cone not the total volume of the cone.

AccipiterQ · Answer

The top surface of the water in the cone is currently  9\pi  square centimeters in area. Top surface area of water =  9\pi=\pi*r^2  -->  r=3  -->  \frac{R}{H}=\frac{r}{h}=\frac{1}{4}

after you solved for r=3, how did you get

\frac{R}{H}=\frac{r}{h}=\frac{1}{4}

Aren't R, H, and h, all unknown?

Bunuel · Answer

We are given that the cone, twice as tall as it is wide, which means that H = 2D --> H = 4R --> R/H = 1/4. Because of similar triangles, the same applies to r and h.

Hope it's clear.

erictwendell · Answer

I really find these problems very tough. Is there any reference to these problems ???

testcracker · Answer

hi

how can you deduce that 2R = H

the cone is twice as tall as it is wide at its greatest width, so

H = 2 x 2R = 4R, which implies

R / 4R = r / h = 1/4, and since, r = 3, h= 12

Isn't that ?

thanks

DarkHorse2019 · Answer

I recommend you to study the basics of 3D Geometries chapter You can find the link of this from the GMAT Club Math book by bb & Bunuel here https://gmatclub.com/forum/math-3-d-geo ... ml#p792331

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