Bosco

What is the cubic yard of 27,000sq ft?     

Assuming you meant 27,000 cubic feet . . .     

There are 27 cubic feet in 1 cubic yard,     

so 1,000 cu yd comprises 27,000 cu ft.     

_.     

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