A square is formed by joining the midpoints of alternate sides of a regular octagon with side length 14 in. Find the shaded area.

98(1 + 2√2) in² ≈ 375 in²

Step-by-step explanation:

Assuming the shaded region is outside of the square and inside of the octagon, we can find the area by subtracting the area of the square from the area of the octagon.

The area of a regular octagon is 2 (1 + √2) s².  We can show this by finding the area of the square outside of the octagon, and subtracting the triangles in the corners:

(s + √2 s)² − 4 (½ (½√2 s)²)

(1 + √2)² s² − 4 (½ (s²/2))

(1 + 2√2 + 2) s² − s²

2 (1 +√2) s²

The diagonal of the inner square is equal to the width of the octagon, (1+√2) s.  So the side length of the square is:

½√2 (1+√2) s

½(2+√2) s

The area of the square is therefore:

(½(2+√2) s)²

¼(2+√2)² s²

¼(4+4√2+2) s²

¼(6+4√2) s²

½(3+2√2) s²

The area of the shaded region is therefore:

2 (1 +√2) s² − ½(3+2√2) s²

½ s² (4 (1 +√2) − (3+2√2))

½ s² (4 + 4√2 − 3 − 2√2)

½ s² (1 + 2√2)

The side length of the octagon is s = 14 in, so the area is:

½ (14 in)² (1 + 2√2)

= 98(1 + 2√2) in²

≈ 375 in²

1. is 4 and 2. is 3

step-by-step explanation:

so if you look at the first one you see that 1 is to 4 and 2 is to 8 and so on, so you can see for every 1 there is a times 4 so if it were 3 then it would be 3*4 and it is 12

for number 2 it is 3 because 1 is to 3 and 2 is to 6 so for every 1 there is 3 so if it were 3 then there would be 3*3 = 9

answer:

payment just for service  

→

$

87.12

payment for service including tax

→

$

94.09

explanation:

service only with 20% tip  

→

$

72.6

(

1

+

20

100

)

=

$

87.12

in the opening of the question the word 'service' is associated only with the pretax value.

the closing part of the question asks: how much did she pay for the  

service

−−−−−−  

so technically the answer is  

→

$

72.6

(

1

+

20

100

)

=

$

87.12

however, the intention of the question provider may have been that your answer is to be  

$

87.12

(

1

+

8

100

)

=

$

94.09

to 2 decimal places.

'

payment just for service  

→

$

87.12

payment for service including tax

→

$

94.09

tony b

mar 21, 2017

answer:

trying to explain it another way

explanation:

service charge  

.

→

$

72.60

20% tip  

20

100

×

$

72.60

=

$

14.52

−−−−−−  

←

add

$

87.12

tax  

8

100

×

$

87.12

=

$

6.97

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add

$

94.09

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