Exercise 18 Page 244

Let's take a look at the given figures.

We will examine how the perimeter of the figures changes when $1$ more octagon is added. We will first find the perimeters of the given figures by counting their sides. Let's make a table that shows the perimeter of the figures $p$ depending the number of octagons $n .$

Number of Octagons $n$	Perimeter $p$
$1$	$8$
$2$	$14$
$3$	$20$

When we examine the rate of change of the variables, we see that the perimeter of the figures increases by

6

as the number of octagons increases by

1 .

In conclusion, we can say that we add

6

to the perimeter of the previous figure to find the perimeter of the next figure.

Alternative Solution

Writing an Equation Representing the Perimeter of the Figures

The perimeter of a regular octagon can be found by multiplying the length of the sides

a

8,

the number of sides.

P = 8 a

However, this formula does not work for adjacent octagons. Therefore, we can also write an equation that represents the perimeter of the adjacent octagons using the table, and our interpretation of it.

P = 6 n

Does the formula work? Let's check it for the first figure!

P = 6 n

SubstituteII

$P = 8$ , $n = 1$

8 = ? 6 (1)

Multiply

8 \neq = 6

As we can see, the formula does not work for

1

octagon. Let's apply the formula to the other figures and try to figure out the missing piece.

Number of Octagons $n$	$P = 6 n$	Perimeter Found by the Formula	Actual Perimeter
$1$	$P = 6 (1)$	$6$	$8$
$2$	$P = 6 (2)$	$12$	$14$
$3$	$P = 6 (3)$	$18$	$20$

The perimeter found by the incomplete formula is always

2

less than the actual perimeter. Therefore, we can complete our formula by adding

2

to the formula.

P = 6 n + 2

Hints

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Alternative Solution