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The figure above shows a side view of the insert

Expert replies
by NandishSS » Sat Jul 30, 2016 5:16 am
Image

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}
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Source: — Problem Solving |

by [email protected] » Sat Jul 30, 2016 10:57 am
Hi NandishSS,

While this question looks complex, it's actually built around some fairly simple Geometry. It might help to break this calculation down into 'pieces' and think about the rules involved in this 3-dimensional shape.

To start, the 'insert' is a rectangle that's been folded in 2 spots. By definition, it has a length and a width; once we figure out those two dimensions, we can figure out its area.

We're meant to assume that the insert 'touches' the sides, so the width of the insert has to match the width of the box. Since the base of the box is an 8-inch square, the width of the insert is also 8 inches.

Next, we'll work on the 'middle' horizontal piece (it's the easiest part) - since it's horizontal, then it has the same length as the box (which is also 8 inches). Thus, that 'middle piece' is an 8x8 square = 64 in^2

The two diagonal pieces are the same length, so once we figure out one, we can double it and get the total area of those 2 'pieces.' You should notice how a bunch of right triangles are formed. The base of each of those triangles is 8 and the height is 6. This is a classic 3/4/5 right triangle that's been doubled to become a 6/8/10. Thus, the two diagonal pieces are 8x10 rectangles = 80 in^2 each.

64+80+80 = 224

Final Answer: B

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by Matt@VeritasPrep » Thu Aug 04, 2016 8:47 pm
If we recognize the sides of 6 and 8 in each smaller rectangle, we see a 6-8-10 right triangle, so we know each hypotenuse is 10.

Unfolding the inserts, they have length 10 + 8 + 10, since each hypotenuse = 10, and the middle line = 8. This gives us a rectangle with length 28.

We know the box is 8x8, so the width of our rectangle is 8.

Multiplying the dimensions, we have area 28 * 8, or 224.
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