Lesson: Volume and Surface Area

Comment on Volume and Surface Area

A rectangular box, with dimensions of 12 inches by 18 inches by 10 inches, contains soup cans. If each can is a cylinder with a radius of 3 inches and a height of 5 inches, what is the maximum number of soup cans that the box can contain?

A. 6
B. 12
C. 15
D. 30
E. 48

For this ques can we solve it by calculating the volume of the rectangular box divided by the surface area of each cans to get the no of cans?
gmat-admin's picture

That strategy won't work, since you're comparing surface areas and volumes, which are entirely different concepts.

Also, I should note that many students will want to find the volume of one can and then divide it into the total volume of the box. However, this strategy only works if we're pouring the liquid soup into the box.

If we're leaving the soup IN the cans, then we must consider the fact that there will be air pockets.

Here's my step-by-step solution: https://gmatclub.com/forum/a-rectangular-box-with-dimensions-of-12-inche...

Cheers,
Brent

Hi Brent,

Can you please answer this question?

A thin conveyor belt 15 feet long is drawn tightly around two circular wheels each 1 foot in diameter. What is the distance, in feet, between the centers of the two wheels?

(A) (15 - π)/2

(B) (5π)/4

(C) 15 - 2π

(D) 15 - π

(E) 2π
gmat-admin's picture

Here are a few different solutions: https://gmatclub.com/forum/as-shown-in-the-figure-above-a-thin-conveyor-...

Please let me know if you'd like me to elaborate on any of them.
Or I can provide a full solution if you wish.

Cheers,
Brent

Hi Brent, in the context of the below question can there be a general statement that says if any container (cylinder, cube, rectangle) is half filled with water, the volume will remain the same if the same shape is rotated to the other side? (https://gmatclub.com/forum/a-closed-cylindrical-tank-contains-36pi-cubic-feet-of-water-134500.html)

A closed cylindrical tank contains 36pi cubic feet of water and is filled to half its capacity. When the tank is placed upright on its circular base on level ground, the height of the water in the tank is 4 feet. When the tank is placed on its side on level ground, what is the height, in feet, of the surface of the water above the ground?
gmat-admin's picture

Question link: https://gmatclub.com/forum/a-closed-cylindrical-tank-contains-36pi-cubic...

If a container is half filled with water, then the volume of water will remain the same, regardless of how the container is positioned (as long as no water spills out).

Does that answer your question? I wasn't 100% sure what you were asking.

Cheers,
Brent

Hi Brent, can the below question be solved using the concept of surface area? Surface area of container - surface area of cylinder = surface area of water with new height?

https://gmatclub.com/forum/a-solid-cylinder-with-radius-3-inches-sits-in-a-cylindrical-234443.html

gmat-admin's picture

Question link: https://gmatclub.com/forum/a-solid-cylinder-with-radius-3-inches-sits-in...

It's not immediately clear to me how that strategy would work.

Can you show me some additional steps?

Cheers,
Brent

Hi Brent,

I couldn't find any lesson on Rectangular Solids and how to calculate their Surface Area, Volume, etc. I was wondering if these are outside the scope of GMAT or whether I am missing something here.

Thanks & Regards,
Abhirup
gmat-admin's picture

Hi Abhirup,

Those concepts are definitely tested on the GMAT, and the above video lesson covers those concepts.

Cheers,
Brent

Hi Brent, facing difficulty interpreting the below question

A crate measures 4 feet by 8 feet by 12 feet on the inside. A stone pillar in the shape of a right circular cylinder must fit into the crate for shipping so that it rests upright when the crate sits on at least one of its six sides. What is the radius, in feet, of the pillar with the largest volume that could still fit in the crate?

2
4
6
8
12
gmat-admin's picture

To help you interpret the problem, here's a graphic showing 2 different ways a cylinder might be placed in a box: https://imgur.com/6SVrKIO

The question you've posted is basically a copy of an official GMAT question (except the dimensions are 6x8x10 instead of 4x8x12).
If you take a look at my full solution (https://gmatclub.com/forum/the-inside-dimensions-of-a-rectangular-wooden...) to that question, I'm pretty sure you'll be able to answer the question you posted.

Cheers,
Brent

https://gmatclub.com/forum/what-is-the-number-of-cans-that-can-be-packed-in-a-certain-143776.html

The explanations mention that we need to know the dimensions of the carton for this to be sufficient. My question is, when do we require the need to use surface area as opposed to volume? If the surface were given instead of volume would that be different?

Fitting X objects into one big Y object seems to be quite common in DS problems.
gmat-admin's picture

Question link: https://gmatclub.com/forum/what-is-the-number-of-cans-that-can-be-packed...

This isn't a very common question type. It's just that most test prep companies have created similar questions that are very similar to this Official Guide question.

I don't think I've ever seen a question of this nature (fitting cylinders into a box) where we are given the cylinder's surface area.
I wouldn't worry about that scenario.

Cheers,
Brent

https://gmatclub.com/forum/what-is-the-volume-of-the-largest-cylinder-that-can-fit-into-a-box-of-275921.html?fl=similar

I'm having trouble visualising these kind of questions. How do I determine which sides would allow me to fit the largest objects inside.

I came up with 160 pie with height of 10 and radius 4
gmat-admin's picture

Question link: https://gmatclub.com/forum/what-is-the-volume-of-the-largest-cylinder-th...

You're right; these kinds of questions require the mental manipulation of various images.

In your solution, the height of the cylinder is 10.
This means, the flat part of the can is lying on the side with dimensions 6 x 8.
If the radius of the can is 4, then the DIAMETER is 8. Since the bottom of the box has dimensions 6 x 8, a can with diameter 8 won't fit in the box.

Cheers,
Brent

https://gmatclub.com/forum/what-is-the-volume-of-a-certain-rectangular-solid-90748.html

What if the dimensions in statement 1 were prime numbers? Would you be able to prove that its sufficient?
gmat-admin's picture

Do you mean statement 1 is as follows:

1) Two adjacent faces of the solid have areas 15 and 24, respectively AND the dimensions of the solid are prime numbers.

This scenario would create an impossible situation. If one side has an area of 24, then the two dimensions of that rectangle must have a product of 24.
However, there is no way to write 24 as the product of two prime numbers.

Cheers,
Brent

No I meant to say that if we suppose statement 1 was:

1) Two adjacent faces of the solid have areas Prime Number X and Prime Number Y respectively.
gmat-admin's picture

Do you mean for statement 1 to NOT have any actual values?

Here are some possible re-wordings for statement 1:

1) Two adjacent faces of the solid have areas PRIME NUMBER X and PRIME NUMBER Y respectively.
This is definitely not sufficient (no values given)

1) Two adjacent faces of the solid have areas 7 and 11 respectively.
This is still not sufficient, since the dimensions COULD be 0.5 x 14 x 22, or they COULD be 1 x 7 x 11

1) Two adjacent faces of the solid have areas 7 and 11 respectively, AND each dimension is an INTEGER value.
SUFFICIENT. Here, the dimensions MUST be 1 x 7 x 11

Does that help?

Cheers,
Brent

Yes that helps. Thanks.

Hi Brent,

Could you provide your solution for this?

https://gmatclub.com/forum/a-tank-is-filled-with-gasoline-to-a-depth-of-exactly-2-feet-the-tank-202262.html

I have trouble understanding how statement 2 is sufficient.

Thanks for your help.

Hi Brent,

https://gmatclub.com/forum/what-is-the-radius-of-the-largest-sphere-that-can-fit-into-a-box-of-di-275651.html

Can you explain how the answer still remains the same if the height is 12?
gmat-admin's picture

Question link: https://gmatclub.com/forum/what-is-the-radius-of-the-largest-sphere-that...

The orientation of the box doesn't matter.
All that matters is that the maximum size of the sphere is restricted by shortest dimension of the box (in this case 8).
This means a sphere with radius 4 (i.e., diameter 8) will touch both sides of the box, which means a sphere with a radius greater than 4 will not fit in the box.

Does that help?

Hi Brent, could you please guide me on solving this question?
https://gmatclub.com/forum/a-square-wooden-plaque-has-a-square-brass-inlay-in-the-center-leaving-89215.html
My approach was:
Brass area = 25 = 5^2;
Total area = (5+2x)^2;
Wooden area = 39 = (5+2x)^2 - 25 => apply a^2-b^2 formula, simplify equation and plug in values. However, I couldn't get to the correct answer.
gmat-admin's picture

We don't actually need to perform any calculations, as you will see in my solution: https://gmatclub.com/forum/a-square-wooden-plaque-has-a-square-brass-inl...

Here's an analogous question:

Joe has two trees in his backyard: A pine tree and a fir tree.
The ratio of the height of the pine tree to the height of the fir tree is 2 : 1.
Which of the following could be the height of the pine tree?
I. 8 meters
II. 10 meters
III. 45 meters

Answer: I, II and III

Brent, is there a simpler way to understand and approach this question?
https://gmatclub.com/forum/the-dimensions-of-a-ream-of-paper-are-8-1-2-inches-by-11-inches-by-294408.html
gmat-admin's picture

I think this approach is probably best: https://gmatclub.com/forum/the-dimensions-of-a-ream-of-paper-are-8-1-2-i...

I can't think of an easier/faster solution.

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