Find the scale factor and ratio of perimeters for a pair of similar octagons with areas 36 ft squared and 49 ft squared.

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03-05-2020
Mathematics

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Find the scale factor and ratio of perimeters for a pair of similar octagons with areas 36 ft squared and 49 ft squared.

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jmonterrozar jmonterrozar · Answer 1 · 2020-05-09T02:34:10+02:00

Answer:

scale factor: 1.16

perimeters: 21.84 ft and 25.36 ft

Step-by-step explanation:

The area for an octagon is as follows:

[tex]A = 2*l^{2}(1+\sqrt{2})\\[/tex]

solved for "l":

[tex]l = \sqrt{\frac{A}{2*(1+\sqrt{2}) } }[/tex]

now we calculate the value of each side:

[tex]l1 = \sqrt{\frac{36}{2*(1+\sqrt{2}) } }[/tex]

l1 = 2.73

[tex]l2 = \sqrt{\frac{49}{2*(1+\sqrt{2}) } }[/tex]

l2 = 3.17

The scale would be:

3.17 / 2.73 = 1.16

And the value of the perimeter of the first would be:

2.73 * 8 = 21.84

and the second:

3.17 * 8 = 25.36

facundo3141592 facundo3141592 · Answer 2 · 2022-04-08T10:57:30+02:00

The scale factor of dilation is k = 1.17, and the ratio of the perimeters is also 1.17

How to get the scale factor?

We know that for an octagon of sidelength S, the area is:

A = 2*S^2*(1 + √2).

In this case, we know that the areas of our octagons are:

36ft^2 = 2*S^2*(1 + √2).

Solving for S we get:

36ft^2/2 = S^2*(1 + √2).

√((18 ft^2)/(1 + √2)) = S = 2.73ft

For the other octagon we have:

49ft^2 = 2*S'^2*(1 + √2)

S' = √(49ft^2)/(2*(1 + √2)) = 3.19 ft

Taking the quotient between the two sidelengths, we get:

3.19/2.73 = 1.17

So the octagon with an area of 49 ft^2 is a dilation of scale factor = 1.17

How to get the perimeters?

Octagons have 8 equal sides, so for the smaller octagon the perimeter is:

P = 8*2.73ft = 21.84ft

For the larger octagon:

P' = 8*3.19 ft = 25.52 ft

Then the ratio of the perimeters is:

P'/P = 25.52ft/21.84ft = 1.17

If you want to learn more about octagons, you can read:

https://brainly.com/question/1592456