Bunuel wrote:

In the scale drawings above, each unit along the x-axis represents 2 meters and each unit along the y-axis represents 1 meter. Which of the rectangles above represents the rectangle with the greatest perimeter?

(A) A

(B) B

(C) C

(D) D

(E) E

Attachment:

2017-12-12_2125.png

Rectangles: maximizing perimeterThere are two geometric properties that help here:

Rectangle with fixed area? A square has the smallest perimeter

Rectangle with fixed perimeter? A square has the largest area

These rectangles have neither fixed areas nor fixed perimeters.

Still, "square-shaped" rectangles will have a smaller perimeter than the long and thin rectangles, except A, because it runs vertically.*

In other words, save time by eliminating B and E immediately.

Eliminate A as well. Each unit is 2 meters wide by 1 meter high.

A lies vertically, the "wrong way" to maximize perimeter on this grid.

Now only C and D remain. C's base is 8*2=16, other side = 1. D's base is 7*2=14, other side 2.

C's area is 16. D's area is 14. From properties above, C will have the larger perimeter.

Just in case:

C measures 8 units of 2 meters long, and 1 unit of 1 meter tall.

L = (8 * 2) = 16

W = (1 * 1) = 1

Perimeter is (2L + 2W): (32 + 2) =

34D measures 7 units of 2 meters long, and 2 units of 1 meter tall

L = (7 * 2) = 14

W = (2 * 1) = 2

Perimeter is (2L + 2W): 28 + 4 =

32Answer C

Brute force Works, too, though it might take more time.

Call each side that runs horizontally "length," and each side that runs vertically, "width."

Each horizontal unit = 2. Each vertical unit = 1

A: 1 by 6

Length = (1 * 2) = 2

Width = (6 * 1) = 6

Perimeter: 2L + 2W = (4 + 12) = 16

B: 4 by 3

Length = (4 * 2) = 8

Width = (3 * 1) = 3

Perimeter: 2L + 2W = (16 + 6) = 22

C: 8 by 1

Length = (8 * 2) = 16

Width = (1 * 1) = 1

Perimeter: 2L + 2W = (32 + 2) = 34

D: 7 by 2

Length = (7 * 2) = 14

Width = (2 * 1) = 2

Perimeter: 2L + 2W = (28 + 4) = 32

E: 3 by 4

Length = (3 * 2) = 6

Width = (4 * 1) = 4

Perimeter: 2L + 2W = (12 + 8) = 20

Answer C

*with a caveat. Sides that run horizontally are measured with 2 per horizontal unit. Sides that run vertically are measured with 1 per vertical unit.

Hence a tall, thin rectangle whose longest side parallels the y-axis will have a perimeter much smaller than a long, thin rectangle whose longest side parallels the x-axis. Compare A and C, for example. A's base is 1 * 2 = 2. C's base is 8 * 2 = 16. Vertical y-units = 1. A would have to be a lot taller than 6 units just to be in the ballpark of C's perimeter.

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