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The figure above shows a side view of the insert and the four components of a box designed to package four lightbulbs. The package is a rectangular box, whose base is a square with a side of 8 inches and whose height is 12 inches. A single rectangular insert is folded twice to form two identical diagonal planes, one at each end, and a horizontal plane in the middle, so that the box is divided into four identical compartments. What is the area of the insert in square inches?
A) 192
B) 224
C) \(64+4\sqrt{52}\)
D) \(64+96\sqrt{2}\)
E) \(64+128\sqrt{2}\)
Attachment:
Figure.jpg [ 4.78 KiB | Viewed 27936 times ]
13 Nov 2017, 04:47
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NandishSS
The figure above shows a side view of the insert and the four components of a box designed to package four lightbulbs. The package is a rectangular box, whose base is a square with a side of 8 inches and whose height is 12 inches. A single rectangular insert is folded twice to form two identical diagonal planes, one at each end, and a horizontal plane in the middle, so that the box is divided into four identical compartments. What is the area of the insert in square inches?
A) 192
B) 224
C) \(64+4\sqrt{52}\)
D) \(64+96\sqrt{2}\)
E) \(64+128\sqrt{2}\)
Attachment:
Figure.jpg
Too much of confusing explanation is given in the question which makes me think that the diagram tells it all. That is pretty much what the first sentence of the question says too.
We see the 8 x 12 rectangular box as shown in the figure.
A single rectangular insert is folded twice to make four identical compartments. We see in the figure that we have 4 identical right triangular compartments. So the insert is just the lightening shape shown in the figure. We need its area. It is actually a rectangle which is folded into this shape. If we open it up, we will get a rectangle and its area will be the total height * Width (which is 8 since the box has a square base of side 8)
Now all we need is the height. Since the compartments are equal, height of each compartment is 6. So each diagonal in the figure is 10 (using multiple of pythagorean triplet 3-4-5).
Total height of the rectangular insert = 10 + 8 + 10 = 28
Its area = 28*8 = 224
29 Mar 2022, 20:56
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tkorzhan1995 I hope the figure attached will help you to understand this question better.
Note: If you could visualize a 3D figure based on the information's given and understand what area we need to find here, solving this question would be easier.
We have a rectangular box, whose base is a square with a side of 8 inches and whose height is 12 inches. A single rectangular sheet ( Yellow shaded region in the attached figure ) is folded twice to form two identical diagonal planes, one at each end, and a horizontal plane in the middle so that the box is divided into four identical compartments. We are asked to find the area of the rectangular insert.
Please refer to the figure attached for the 3D -view and we are asked to find the yellow shaded region in the figure.
3D_FIGURE.JPG [ 38.36 KiB | Viewed 8738 times ]
Let's cut the entire rectangular insert into three parts: Sheets 1,2 and 3 as marked in the figure. The width of each sheet will be the same i.e 8 inches.
Since the box is divided into 4 identical compartments, the height of each compartment will be 12/2 = 6 inches.
The length of rectangular sheet 1 will be the hypotenuse of the right-angled triangle formed as shown in the figure with a base of 8 inches and a height of 6 inches.
Length of sheet 1 = Sq.root(\(8^2 + 6^2\)) = 10 inches
Area of Sheet 1 = 10 * 8 = 80 sq. inches
It's given in the question that Sheet 1 and Sheet 3 are two identical diagonal planes, hence their area will also be the same i.e 80 sq. inches.
Sheet 2 will be a square of base 8 cm, hence the area of sheet 2 = 8 * 8 = 64 sq. inches
Area of rectangular insert = Area of Sheet 1 + Area of sheet 2 + Area of sheet 3= 80 + 64 + 80 = 224 sq. inches.
Option B is the correct answer.
Thanks,
Clifin J Francis,
GMAT Mentor
Note: If you could visualize a 3D figure based on the information's given and understand what area we need to find here, solving this question would be easier.
We have a rectangular box, whose base is a square with a side of 8 inches and whose height is 12 inches. A single rectangular sheet ( Yellow shaded region in the attached figure ) is folded twice to form two identical diagonal planes, one at each end, and a horizontal plane in the middle so that the box is divided into four identical compartments. We are asked to find the area of the rectangular insert.
Please refer to the figure attached for the 3D -view and we are asked to find the yellow shaded region in the figure.
Attachment:
3D_FIGURE.JPG [ 38.36 KiB | Viewed 8738 times ]
Let's cut the entire rectangular insert into three parts: Sheets 1,2 and 3 as marked in the figure. The width of each sheet will be the same i.e 8 inches.
Since the box is divided into 4 identical compartments, the height of each compartment will be 12/2 = 6 inches.
The length of rectangular sheet 1 will be the hypotenuse of the right-angled triangle formed as shown in the figure with a base of 8 inches and a height of 6 inches.
Length of sheet 1 = Sq.root(\(8^2 + 6^2\)) = 10 inches
Area of Sheet 1 = 10 * 8 = 80 sq. inches
It's given in the question that Sheet 1 and Sheet 3 are two identical diagonal planes, hence their area will also be the same i.e 80 sq. inches.
Sheet 2 will be a square of base 8 cm, hence the area of sheet 2 = 8 * 8 = 64 sq. inches
Area of rectangular insert = Area of Sheet 1 + Area of sheet 2 + Area of sheet 3= 80 + 64 + 80 = 224 sq. inches.
Option B is the correct answer.
Thanks,
Clifin J Francis,
GMAT Mentor
