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Powers of 4

4^1 =  4

4^2 = 16

4^3 = 64

4^4 = 256

In general   4^(2n)  ends in 6

So...the  units digit of 1^1 +  2^2 + 3^4 + 4^8  =

1 + 4 + 1 + 6  =   2

Rhombus ABCD has perimeter 148, and one of its diagonals has length 20. How long is the other diagonal?     

The Parallelogram Law states that the sum of the squares of     

the diagonals equals the sum of the squares of the four sides.     

A rhombus is simply a parallelogram with all four sides equal.     

d₁² + d₂²  =  4 s²     

d₁² + 20²  = 4 • 37²     

          d₁²  =  (4 • 1369) – 400     

          d₁²  =  5476 – 400     

          d₁²  =  5076     

          d₁¹  =  sqrt(5076)     

          d₁¹  =  71.25     

_.     

Linda, Miah, and Noah are all wearing hats with numbers on them. Each person adds the two numbers on the other two hats, and they come up with the totals 11, 17, and 28. What is the largest number that is on a hat?     

a + b = 11     —>  b = 11 – a     

a + c = 17     —>  c = 17 – a     

b + c = 28     sub the above   —>  (11 – a) + (17 – a)  =  28     

                                                                       28 – 2a  =  28     

                                                                            – 2a  =   0     

                                                                                 a  =   0     

                                                    0 + b = 11     —>   b  =  11     

                                                    0 + c = 17     —>   c  =  17     

_.     

Insert three fractions between 1/4 and 1/3  so the five fractions form an arithmetic sequence.     

What is the sum, expressed as a common fraction, of these three new fractions?     

We need a common denominator.  Let's see how 24 will work.     

24 no funciona.  There's not enough room between 1/4 & 1/3.     

What about 48?     12/48   13/48   14/48   15/48   16/48     

48 worked good, so           13/48 + 14/48 + 15/48  =  42/48  =  7/8     

_.     

The arithmetic mean of 23, 16, 30, 33 and two other numbers that differ by 2 is 27. If the smallest of these 6 numbers is discarded, find the arithmetic mean of the remaining 5 numbers.     

                                    [ 16 + 23 + 30 + 33 + x + (x+2) ] / 6  =  27     

                                                               [ 102 + 2x + 2 ] / 6  =  27     

                                                                    [ 102 + 2x + 2 ]  =  162     

                                                                                        2x  =  162 – 104  =  58     

                                                                                          x  =  29   &   x+2 = 31     

                                                                           New Mean  =  (16 + 23 + 30 + 33 + 31) / 5     

                                                                                               =  133 / 5     

                                                                                               =  26.6     

_.     

Heres a question that our teacher assigned over the weekend: carlos stands 9 meters from the center of a merry-go round that is spinning at constant speed, and clarita stands 6 meters from the center. If Carlos speed is 3m/s what is claritas speed in meters per second      

The circle that Clarita travels is π 2 r_clarita     

The circle that Carlos travels is π 2 r_carlos     

Divide them to find the ratio of Clarita's circle to Carlos' circle    

After canceling the 2 π , we find the ratio is r_clarita/r_calros = 6/9     

Since they both make it all the way around in the same amount of time,     

and Clarita's distance is 2/3 that of Carlos' distance, Clarita is traveling     

2/3 as fast as Carlos.  Therefore Clarita's speed is 2/3 x 3 m/s = 2 m/s.     

_.     

Two sides of a parallelogram are 30 and 20.     

One diagonal has a length of 40.     

What is the length of the other diagonal?      

 The Parallelogram Law states that the sum of the squares of     

 the diagonals equals the sum of the squares of the four sides.     

 (d₁² + d₂²)  =  2 • (a² + b²)     

 d₁² + 40²  =  2 • (30² + 20²)     

 d₁² + 1600  =  2 • (900 + 400)     

 d₁²  =  1800 + 800 – 1600     

 d₁²  =  1000     

 d₁  =  sqrt(1000)     

 d₁  =  10 • sqrt(10)     —>>  about 31.62     

_.     

Find all positive integer values of c such that the equation x^2 - 11x + c = 0 only has roots that are real and rational. Express them in decreasing order, separated by commas.     

x² – 11x + c = 0     

(x – 5)(x –   6) = 0     x = 5, x =   6     —>>  c = 30     

(x – 4)(x –   7) = 0     x = 4, x =   7     —>>  c = 28     

(x – 3)(x –   8) = 0     x = 3, x =   8     —>>  c = 24     

(x – 2)(x –   9) = 0     x = 2, x =   9     —>>  c = 18     

(x – 1)(x – 10) = 0     x = 1, x = 10     —>>  c = 10     

     Write 214 feet as a multiplication expression using the unit, 1 foot, as a factor.     

     214 x 1 ft  =  214 ft     

_.     

A circle is centered at O.  Find the radius of the circle.     

A = π r²     

A  =  9 π ft²     

π r²  =  9 π ft²    divide both sides of this by π     

    r²  =  9 ft²     

    r  =  3 ft     

Assuming a right triangle

Let the legs =  AC =  sqrt 7    and  BC =   sqrt 3

The hypotenuse = 

sqrt [ (sqrt .7)^2  + (sqrt .3)^2]   =  sqrt [ 7 + 3]  = sqrt 10  = AB

So....noting that cos B  = sinA

cos^2 B + cos^2 A  =

sin^2 A + cos^2A  = 1

(sqrt (3/10)^2  +(sqrt (7/10)^2  =  1

3/10  +  7/10  =  1

Then  angle C  is  opposite the hypotenuse  so its measure = 90°

And cos C =   cos (90°)  =  0

Angle DAB  =90

So angle DAE  = 90 / 3 =  30

And DAE  is a 30- 60 -90 right triangle

DE is  opposite the 30 degree angle

And AD is opposite the 60 degree angle

So AD =  6sqrt 3  =  BC

So

BF  = BC - FC   =   6sqrt 3  - 1  =   5sqrt 3

semi-circle on top  right with radius =  4  =  16pi/2  = 8pi  km^2

area of rectangle on left  side of  semi-circle = 8 * 5  = 40 km^2

area of small rectangle in the middle  = 2 * 3 = 6 km^2

area of rectangle at bottom = 5 * 8  = 40 km^2

Total area  = [  8pi + 40 + 6 + 40 ] km^2  =

[ 86 + 8pi] km^2 ≈  111.13 km^2

Tom's mathematics test had 60 questions: 10 arithmetic questions, 30 algebra questions and 20 geometry questions. Although he correctly answered 70% of the arithmetic questions, 40% of the algebra questions and 50% of the geometry questions, he did not pass the test because he got fewer than 60% of the questions right. How many more questions would he have needed to answer correctly to earn a passing grade of 60%?     

10 • 70%  =    7 correct     

30 • 40%  =  12 correct     

20 • 50%  =  10 correct     

       total       29 correct     

60 • 60%  =  36 to pass     

36 – 29  =  7     Tom needed 7 more correct answers to pass     

_.     

   Which of these can factor using difference of two squares ?     

   4.  144x² – 49y²  =  (12x + 7y)(12x – 7y)     

_.     

In right triangle XYZ with YXZ = 90 degrees, we have XY = 24 and YZ = 28. Find tan Y.     

                  XZ²  =  28² – 24²     

                  XZ²  =  784 – 576     

                  XZ²  =  208     

                  XZ    =  sqrt(208)     

                  XZ    =  14.4222     

                                                tangent Y  =  XZ / XY     

                                                tangent Y  =  14.4222 / 24     

                                                tangent Y  =  0.6009     

_.     

 MS. Moua has 20 students in her first-grade class.     

 How many ways can she choose 5 of them to ride     

 in her van for a class field trip?      

         ₂₀C₅  =  15,504     

_.     

In the single-elimination NCAA Tournament, 64 teams compete in 6 bracketed rounds. How many games are there in total?     

64 teams would play 32 games ——>  32 teams go home, 32 teams play again     

32 teams would play 16 games ——>  16 teams go home, 16 teams play again     

16 teams would play   8 games ——>    8 teams go home,   8 teams play again     

  8 teams would play   4 games ——>    4 teams go home,   4 teams play again     

  4 teams would play   2 games ——>    2 teams go home,   2 teams play again      

  2 teams would play   1 game   ——>    1 team goes home, 1 team takes the trophy home     

                                   –––––––     

add up all games       63 games      

_.     

In a box there are red and blue balls. If you select a handful of them with eyes closed, you have to grab at least 5 of them to make sure at least one of them is red and you have to grab at least 10 of them to make sure both colors appear among the balls selected. How many balls are there in the box?     

"you have to grab at least 5 of them to make sure at least one of them is red"     

This tells me that there are four blue balls.     

"you have to grab at least 10 of them to make sure both colors appear among the balls selected"     

This tells me that there are nine red balls.     

4 blue + 9 red  =  13 balls total     

_.