Afriend gives you a puppy for your birthday and offers to buy fencing for a dog run. What shape gives your puppy more area to run: a
rectangle whose width is twice the length, a regular hexagon, or a square? -
A
rectangle
B. regular hexagon
C. square

B. regular hexagon

Step-by-step explanation:

Let's say he gives you x length of fencing.

Rectangle

The rectangle has length L and width 2L.

The perimeter is L + 2L + L + 2L

L + 2L + L + 2L = x

6L = x

L = x/6

2L = 2x/6 = x/3

The area of the rectangle is

A = length * width

A = x/6 * x/3 = x^2/18

A = 0.0556x^2

For the rectangle, the area is x^2/18.

Square

The length of each side is L.

The perimeter is 4L.

4L = x

L = x/4

The area of the square is

A = side^2

A = (x/4)^2 = x^2/16

A = 0.0625x^2

For the square, the area is x^2/16.

Regular hexagon

The length of fencing is x.

The regular hexagon has 6 congruent sides.

The length of each side is L.

The perimeter is 6L

6L = x

L = x/6

The length of each side is x/6.

A hexagon can be divided into 6 congruent equilateral triangles.

Each triangle can be divided into 2 congruent right triangles by drawing an altitude from the vertex at the center of the triangle to the midpoint of the side of the hexagon. Each right triangle is a 30-60-90 triangle. Half of the side of the hexagon measures x/12. By the ratios of the lengths of sides of a 30-60-90 triangle, the other leg has length x*sqrt(3)/12. The area of the hexagon is 6 times the area of one equilateral triangle.

area = 6 * area of equilateral triangle

area = 6 * bh/2

area = [6 * x/6 * x * sqrt(3)/12]/2

area = x^2*sqrt(3)/24

A = 0.0722x^2

The approximate areas are:

rectangle: 0.0556x^2

square: 0.0625x^2

hexagon: 0.0722x^2

The largest area is that of the regular hexagon.