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805+ Level|   Geometry|      
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Bunuel
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Right circular cylinders with the radius of 7 cm and height of 30 cm 05 May 2020, 05:32
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Right circular cylinders with the radius of 7 cm and height of 30 cm are to be placed in a box with the dimensions 76 cm by 46 cm by 45 cm. What is the maximum number of cylinders that can be placed in the box?

A. 15
B. 17
C. 20
D. 21
E. 22


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D
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Right circular cylinders with the radius of 7 cm and height of 30 cm 05 May 2020, 06:26
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Right circular cylinders with the radius of 7 cm and height of 30 cm are to be placed in a box with the dimensions 76 cm by 46 cm by 45 cm. What is the maximum number of cylinders that can be placed in the box?

A. 15
B. 17
C. 20
D. 21
E. 22

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img2.png [ 23.62 KiB | Viewed 4000 times ]
If we place a cylinder horizontally we can keep two more on top of it. in total these three cylinders will take 42 cms in height.
As the width is 46 we can keep three cylinders on one layer.
These are basically stacked as \(3\times3\) and we can have two such stacks.
In total these will be 9 + 9 = 18 cylinders placed horizontally
As these stacks have covered 60 cms in length we have another 16 cms which can hold 1 cylinder placed vertically and three such vertical cylinders placed next to each other.
One last cylinder horizontally on top of vertical cylinders.
all packed now!!
total cylinders = 18 + 3 + 1= 22
Ans: E
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Right circular cylinders with the radius of 7 cm and height of 30 cm 05 May 2020, 06:58
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Bunuel
Right circular cylinders with the radius of 7 cm and height of 30 cm are to be placed in a box with the dimensions 76 cm by 46 cm by 45 cm. What is the maximum number of cylinders that can be placed in the box?

A. 15
B. 17
C. 20
D. 21
E. 22


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Diameter = 2*7 = 14 cm,
Height = 30

i.e. 3 dimensions of cylinder are 14-14-30

i.e. 3 dimensions of Box are 76-45-46

Let's Count the number of upright Cylinders in Box = [76/30]*[45/14]*[46/14] = 2*3*3 = 18

Now, 76 is 16 greater than 60 therefore 16 cm is left unused on this dimension

i.e. Unused Volume of box has dimensions 15*46*45

Number cylinders to be kept in horizontal position while height of cylinder 30 is along 46

[15/14]*[45/30]*[45/14] = 1*1*3 = 3 Cylinder

On Dimension we still have 45-30 = 15 left unused

Now, more cylinder = [15/14]*[15/14]*[46/30] = 1*1*1 = 1

Total Cylinders = 18+3+1 = 22

Answer: Option E
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Right circular cylinders with the radius of 7 cm and height of 30 cm 09 May 2020, 06:25
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Bunuel
Right circular cylinders with the radius of 7 cm and height of 30 cm are to be placed in a box with the dimensions 76 cm by 46 cm by 45 cm. What is the maximum number of cylinders that can be placed in the box?

A. 15
B. 17
C. 20
D. 21
E. 22


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Notice that placing cylinders within the box renders some space between the cylinders unusable; therefore, a cylinder will actually take as much space as a 14 x 14 x 30 box. For simplicity, we can think of the cylinders as 14 x 14 x 30 boxes instead.

Placing cylinders on the 46 x 45 base, we will be able to fit 46/14 = 3 (ignore the remainder) cylinders along the side of length 46 and 45/14 = 3 (ignore the remainder) cylinders along the side of length 45. Thus, 3 x 3 = 9 cylinders can be placed on the bottom of the box. Since 76/30 = 2 (ignore the remainder), we can stack two layers of cylinders, for a total of 9 x 2 = 18 cylinders. So far, we used 60 cm of the 76 cm height of the box, with 76 - 60 = 16 cm of height unused. So, we can place more cylinders in the 45 x 46 x 16 region of the box.

Now, we will follow the same procedure with dimensions 45 x 46 x 16. Notice that we cannot use the base of 45 x 46 anymore, since the height of 30 cm will exceed the maximum height of 16 cm. Thus, let’s use the base 45 x 16. We can fit 45/14 = 3 (ignore the remainder) cylinders along the side of length 45 and 16/14 = 1 (ignore the remainder) cylinder along the side of length 16. This makes it possible to fit 3 x 1 = 3 more cylinders and gives us 46 - 30 = 16 cm of unused height.

Finally, we are left with an unused space of dimensions 45 x 16 x 16. We can only fit one more cylinder in this space.

In total, it is possible to fit 18 + 3 + 1 = 22 cylinders into the box.

Answer: E
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Right circular cylinders with the radius of 7 cm and height of 30 cm 08 Apr 2021, 20:52
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Bunuel
Right circular cylinders with the radius of 7 cm and height of 30 cm are to be placed in a box with the dimensions 76 cm by 46 cm by 45 cm. What is the maximum number of cylinders that can be placed in the box?

A. 15
B. 17
C. 20
D. 21
E. 22



Notice that placing cylinders within the box renders some space between the cylinders unusable; therefore, a cylinder will actually take as much space as a 14 x 14 x 30 box. For simplicity, we can think of the cylinders as 14 x 14 x 30 boxes instead.

Placing cylinders on the 46 x 45 base, we will be able to fit 46/14 = 3 (ignore the remainder) cylinders along the side of length 46 and 45/14 = 3 (ignore the remainder) cylinders along the side of length 45. Thus, 3 x 3 = 9 cylinders can be placed on the bottom of the box. Since 76/30 = 2 (ignore the remainder), we can stack two layers of cylinders, for a total of 9 x 2 = 18 cylinders. So far, we used 60 cm of the 76 cm height of the box, with 76 - 60 = 16 cm of height unused. So, we can place more cylinders in the 45 x 46 x 16 region of the box.

Now, we will follow the same procedure with dimensions 45 x 46 x 16. Notice that we cannot use the base of 45 x 46 anymore, since the height of 30 cm will exceed the maximum height of 16 cm. Thus, let’s use the base 45 x 16. We can fit 45/14 = 3 (ignore the remainder) cylinders along the side of length 45 and 16/14 = 1 (ignore the remainder) cylinder along the side of length 16. This makes it possible to fit 3 x 1 = 3 more cylinders and gives us 46 - 30 = 16 cm of unused height.

Finally, we are left with an unused space of dimensions 45 x 16 x 16. We can only fit one more cylinder in this space.

In total, it is possible to fit 18 + 3 + 1 = 22 cylinders into the box.

Answer: E
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Hi ScottTargetTestPrep, can you maybe sketch this out with labels? I am having a really hard time visualizing this.

Edit: I think I get it now.

Question though, could we have done what you did but switched the position of 46 and 45?
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Right circular cylinders with the radius of 7 cm and height of 30 cm 19 Aug 2023, 02:11
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