geno3141

Since there are ten distinct integers, the median must be the average of the highest number (in the set) below 12 and the smallest number above 12 (in the set).

You will want the four smallest numbers below 12:      1, 2, 3, and 4.

You also want the five smallest numbers above 12:     13, 14, 15, 16, and 17.

To get the mean of the highest number below 12 and the smallest number above 12, these two middle numbers must be 11 and 13.

So, the numbers are:  1, 2, 3, 4, 11, 13, 14, 15, 16, and 17.

You will need to find the mean of these numbers to get your answer.

Call the first numerator:  x

Then, the first denominator is:  x - 1

Decreasing each numerator and denominator by 1 gives the following expression:

(x)/(x - 1)  ·  (x - 1)/(x - 2)  ·  (x - 2)/(x - 3)  ·  (x - 3)/(x - 4)  ·  ...  ·  (x - 19)/(x - 20)

Cancelling all the factors that we can cancel leaves us with:  x / (x - 20)

Therefore:  x / (x - 20)  =  3     --->     x  =  3(x - 20)     --->     x  =  3x - 60

                                                                                             -2x  =  -60

                                                                                                x  =  30

The first fraction was:  30 / 29

Let the five consecutive integers be:  x,  x + 1,  x + 2,  x + 3,  x + 4

Their sum is:  5x + 10.

Since this sum is 14 more than the greatest:  5x + 10  =  14 + (x + 4)

                                                                         5x + 10  =  18 + x

                                                                                  4x  =  8

                                                                                    x  =  2

Similar figures have areas that go by the square of the original sides.

So, if the two dimensions are each multiplied by 3, the area of the new figure will be 3² or 9 times the area of the original figure.

Since the new triangle is 54 square feet, the original triangle will be 54 / 9 = 6 square feet.

To check:  assume that the original triangle had a base of 3 feet and a height of 2 feet, resulting in 6 square feet.

Then, the new triangle will have a base of 3 x 3 = 9 feet and a height of 2 x 3 = 6 feet, resulting in 54 square feet.

If the formula is:  y  =  16 / (5 + 3x)  to find the inverse:

1)  interchange the x and y values:                    x  =  16 / (5 + 3y)

2)  Solve this equation for y:                  x(5 + 3y)  =  16

                                                                   5 + 3y  =  16/x

                                                                         3y  =  16/x - 5

                                                                           y  =  ( 16/x - 5 ) / 3

Now, to finish:

3)  Substitute 2 for x and reduce:

4)  Square that answer:

5)  Invert that answer:

The formula for the volume of a cylinder is:  Volume  =  pi · radius² · height

Entering the known values:                            392·pi  =  pi · radius² · height

Dividing both sides by pi:                                    392  =  radius² · 8

Dividing both sides by 8:                                       49  =  radius² 

Taking the square root of both sides:                      7  =  radius

You can picture these cities as all being on the same highway.

CC - YT - MT - LV

CC to YT = 37

YT to MT = 6

MT to LV = 8

This is a total of 51 miles, the same distance as from CC to LV.

This means that CC to MY is 43 miles.

Since it is a geometric sequence, ask yourself "What number times 1 will give you 2?"   --->  2

Again:  "What number times 2 will give you 4?"   --->   again, the answer is 2.

So, this is your common ratio; you can find all the rest of the terms by multiplying each preceding term by 2.

The common ratio is  sqrt(2).

The  sqrt(2)  can be written as  2^1/2.

A formula for the n^th term of a geometic sequence is:  a_n  =  a₁ · r^{n - 1}.

Filling in the known values:                                          192  =  3 · (2^1/2)^{n - 1}

Dividing by 3:                                                                 64  =   (2^1/2)^{n - 1}

Rewriting 64:                                                                  2⁶  =  (2^1/2)^{n - 1}

Looking at only the exponents:                                        6  =  (1/2)(n - 1)

                                                                                       12  =  n - 1

                                                                                         n  =  13

As you count from 1 to 13 and then back down again, you get this set of numbers:

1-2-3-4-5-6-7-8-9-10-11-12-13-12-11-10-9-8-7-6-5-4-3-2

Then your next set of numbers repeat the above list

1-2-3-4- etc.

There are 24 numbers in each list.

So, how many sets of 24 are in 1234?

Dividing 1234 by 24, we get a quotient of 51 and a remainder of 10.

The quotient of 51 tells me that there are 51 groups of 24 numbers (which I really don't care about)

and the remainder of 10 tells me how far to keep counting (I care about this!)

So what number is the tenth number in the list

     1-2-3-4-5-6-7-8-9-10-11-12-13-12-11-10-9-8-7-6-5-4-3-2 ?

The tenth number in the list is my answer ...