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# The inside dimensions of a rectangular wooden box are 6 inches by 8 in

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Joined: 02 Sep 2009
Posts: 52971
The inside dimensions of a rectangular wooden box are 6 inches by 8 in  [#permalink]

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04 Dec 2017, 21:57
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Difficulty:

95% (hard)

Question Stats:

45% (02:04) correct 55% (01:55) wrong based on 64 sessions

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The inside dimensions of a rectangular wooden box are 6 inches by 8 inches by 10 inches. A cylindrical cannister is to be placed inside the box so that it stands upright when the closed box rests on one of its six faces. Of all such cannisters that could be used, what is the radius, in inches, of one that has the maximum volume?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 8

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Concentration: Healthcare, Finance
Re: The inside dimensions of a rectangular wooden box are 6 inches by 8 in  [#permalink]

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Updated on: 05 Dec 2017, 08:17
Bunuel wrote:
The inside dimensions of a rectangular wooden box are 6 inches by 8 inches by 10 inches. A cylindrical cannister is to be placed inside the box so that it stands upright when the closed box rests on one of its six faces. Of all such cannisters that could be used, what is the radius, in inches, of one that has the maximum volume?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 8

Let's assume that the bottom side of the box has dimensions 8*10 when the canister is placed in it. Obviously, the diameter can't be more than eight so the radius is 4 Height is 6
$$V = \pi * r^2 * h = 16\pi * 6 = 96\pi$$

Let's assume that the bottom side of the box has dimensions 8*6 when the canister is placed in it. Obviously, the diameter can't be more than 6 so the radius is 3 Height is 10
$$V = \pi * r^2 * h = 9\pi * 10 = 90\pi$$

Originally posted by Vorovski on 04 Dec 2017, 23:31.
Last edited by Vorovski on 05 Dec 2017, 08:17, edited 2 times in total.
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Posts: 151
Re: The inside dimensions of a rectangular wooden box are 6 inches by 8 in  [#permalink]

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Updated on: 07 Dec 2017, 15:41
Bunuel wrote:
The inside dimensions of a rectangular wooden box are 6 inches by 8 inches by 10 inches. A cylindrical cannister is to be placed inside the box so that it stands upright when the closed box rests on one of its six faces. Of all such cannisters that could be used, what is the radius, in inches, of one that has the maximum volume?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 8

Volume of a cylinder: $$pi*r^2*h$$
Radius = $$\frac{Diameter}{2}$$

The maximum diameter of a cylinder that fits inside the rectangular box is restricted to the length and width of the box.
So, if we have a length =8, width = 10 and height = 6, then max. diameter = 8 and radius = 4;
And, the volume of the cylinder: $$pi*4^2*6=pi*96$$

Let's try another, with length =8, width = 6 and height = 10, then max. diameter = 6 and radius = 3;
And, the volume of the cylinder: $$pi*3^2*10=pi*90$$

Largest radius we can have is 4.

Originally posted by exc4libur on 05 Dec 2017, 04:35.
Last edited by exc4libur on 07 Dec 2017, 15:41, edited 1 time in total.
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Re: The inside dimensions of a rectangular wooden box are 6 inches by 8 in  [#permalink]

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05 Dec 2017, 06:34
1
exc4libur wrote:
Bunuel wrote:
The inside dimensions of a rectangular wooden box are 6 inches by 8 inches by 10 inches. A cylindrical cannister is to be placed inside the box so that it stands upright when the closed box rests on one of its six faces. Of all such cannisters that could be used, what is the radius, in inches, of one that has the maximum volume?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 8

Volume of a cylinder: $$pi*r^2*h$$

Let's test the wooden box dimensions for the bottom and the height.
If we have a bottom of 6 and a height of 10 then the radius would be 3 and volume of cylinder = $$pi*3^2*10 = pi*90$$
If we have a bottom of 8 and a height of 10 then the radius would be 4 and volume of cylinder = $$pi*4^2*10 = pi*160$$
If we have a bottom of 10 and a height of 8 then the radius would be 5 and volume of cylinder = $$pi*5^2*8 = pi*200$$

So the maximum volume of the cylinder would be $$pi*200$$ with a radius of 5.

Might i check if you have considered how the cylinder could fit in?

For example, the length of the bottom is 10 and the height is 8, that means the width the bottom is 6. How do you fit in a cylinder with a radius of 5 (i.e., diameter = 10) while the width is only 6?

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Joined: 27 May 2012
Posts: 683
Re: The inside dimensions of a rectangular wooden box are 6 inches by 8 in  [#permalink]

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25 Oct 2018, 06:12
Bunuel wrote:
The inside dimensions of a rectangular wooden box are 6 inches by 8 inches by 10 inches. A cylindrical cannister is to be placed inside the box so that it stands upright when the closed box rests on one of its six faces. Of all such cannisters that could be used, what is the radius, in inches, of one that has the maximum volume?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 8

Dear Moderator,
Please check the OA for this one, seems as though the answer should be B.
The shortest side of the base determines the diameter and hence the radius of the cannister.
Of all the bases , when the base is 8 one side and 10 other side , and the height 6 , we get the cannister of the max volume. Hence Diameter is 8 and radius is 4.
Hope you will look into this . Thank you.
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- Stne

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Joined: 02 Sep 2009
Posts: 52971
Re: The inside dimensions of a rectangular wooden box are 6 inches by 8 in  [#permalink]

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25 Oct 2018, 06:28
stne wrote:
Bunuel wrote:
The inside dimensions of a rectangular wooden box are 6 inches by 8 inches by 10 inches. A cylindrical cannister is to be placed inside the box so that it stands upright when the closed box rests on one of its six faces. Of all such cannisters that could be used, what is the radius, in inches, of one that has the maximum volume?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 8

Dear Moderator,
Please check the OA for this one, seems as though the answer should be B.
The shortest side of the base determines the diameter and hence the radius of the cannister.
Of all the bases , when the base is 8 one side and 10 other side , and the height 6 , we get the cannister of the max volume. Hence Diameter is 8 and radius is 4.
Hope you will look into this . Thank you.

_______________
Edited. Thank you.
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Re: The inside dimensions of a rectangular wooden box are 6 inches by 8 in  [#permalink]

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28 Oct 2018, 06:08
volume of Cuboid = 480
To get placed inside the cuboid
Volume of cuboid= volume of cylinder
pi r^2 h = 480

Concept of solving the unknown
To stand upright, the cylinder canister should have a height equal to the cuboid largest side (i.e= 10)

solving for above eq.
r~4

(All dimensions are in inches and volume inches^3)
Re: The inside dimensions of a rectangular wooden box are 6 inches by 8 in   [#permalink] 28 Oct 2018, 06:08
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