The diagram below shows twelve 30-60-90 triangles placed in a circle so that the hypotenuse of each triangle coincides with the longer leg of the next triangle. the fourth and last triangle in this diagram are shaded. the ratio of the perimeters of these two triangles can be written as m/n where m and n are relatively prime positive integers. find m + n

m+n = 337

Step-by-step explanation:

Lets start from the first triangle, It is given to be as 30-60-90.

The hypotenous of first rectangle be 2x, then other sides are by default x and

, using laws of trigonometry.

(sin(30) = 0.5  and cos(30) = )

The perimeter of first triangle is ,

=

Now, for second triangle, the longer leg is 2x, and similarly again,

other 2 sides are .

Again the perimeter of triangle comes out as,

=

Thus, the repeating pattern is identified. The consecutive perimeters differ by, multiplying by factor

Thus, we can say that perimeter of 4th triangle is,

= , where X is the repeating constant.

And of 12th triangle is,

= ,

Evaluating the above ratio, we get,

=

Thus, m =256 and n=81.

Thus, m+n = 256+81 = 337.

9,878,813

Step-by-step explanation:

First, make 0.6% into a decimal

0.6 / 100 = 0.006  

Then, multiply the number of people living in Michigan in 2000 by the number of people decreased (the percentage decrease) in 2010

9,938,444 x 0.006 = 59630.664

Last, subtract the percentage decrease by the original population in 2000

9,938,444 – 59630.664 = 9,878,813

In 2010 the approximate population in Michigan was about 9,878,813 (to the nearest whole person)