Bosco

Two trains started moving at the same time, train A from Boston to New York, and train B from New York to Boston.  The distance between the train stations in New York and Boston is 240 miles.  The average speed of train A is 60 mph, which is 3/4 of the average speed of train B.  How soon will the trains meet?     

Train A  =  60 mph     

Train B  =  80 mph     

Combined speed  =  140 mi / hr     

Distance = Rate • Time     ——>>     Time  =  Distance / Rate     

                                                           Time  =  (240 mi) / (140 mi / hr)     

                                                           Time  =  1.714 hr     

0.714 hr       ——>>     (60 min / hr) • ( 0.714 hr)  =  42.84 min     

Round 42.84 to 43       Trains will meet 1 hr & 43 min after they left     

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Rhombus ABCD has perimeter 148, and one of its diagonals has length 28. what is the area of ABCD?     

call one diagonal p and the other diagonal q     

by Pythagoras,    (side_RHOMBUS)²  =  ( half p )² + ( half q )²     

You can do this because     

The diagonals of a rhombus intersect at a 90^o angle.     

The diagonals of a rhombus bisect each other.     

                                     37²  =  14²  + ( half q )²     

                                   1369  =  196  + ( half q )²     

                            ( half q )²  =  1173     

                            ( half q )    =  34.25     

                                          q  =  68.5     

             It was given that   p  =  28     

                              A_rhombus  =  (p • q) / 2     

                              A_rhombus  =  (28 • 68.5) / 2  =  1918 / 2     

                                          A  =  959     

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In a pet shop, there are 3 hamsters for every 2 guinea pigs, and there are 2 giant cloud rats for every 8 guinea pigs. If N is the total number of hamsters, guinea pigs, and giant cloud rats, and N > 0, then what is the smallest possible value of N?     

2 giant cloud rats for every 8 guinea pigs       

this is a proportion so it can be reduced to     

1 giant cloud rat  for every 4 guinea pigs     

1 rat per 4 pigs                                        1 rat     

                                                                4 pigs     

3 hamsters per 2 pigs     

6 hamsters per 4 pigs                             6 hamsters     

add them up                         least N  =  11 animals     

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  Check the multiples of (6 x n.5) then see what works with 8.     

  when  6 x 1.5 =   9     then      9/8 —> remainder 1     

  when  6 x 2.5 = 15     then    15/8 —> remainder 7     

  when  6 x 3.5 = 21     then    21/8 —> remainder 5     

  when  6 x 4.5 = 27     then    27/8 —> remainder 3     

  Least children = 27     

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x² + 3x – 4    =  (x – 1)(x + 4)     

x² + 9x – 10  =  (x – 1)(x + 10)     

When x=1, both (x² + 3x – 4) and (x² + 9x – 10) equal zero     

So, when x=1, (x² + 3x – 4) + (x² + 9x – 10) = 0 + 0  which     

puts a zero in the denominator, and you can't divide by zero.   

Therefore, when x = 1 the fraction is 1/0 which is undefined.       

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A cube has side length . Its vertices are alternately colored black and purple, as shown below. What is the volume of the tetrahedron whose corners are the purple vertices of the cube? (A tetrahedron is a pyramid with a triangular base.)     

"A cube has side length ."  Hard to solve a problem without that piece of information.     

Let's assign our own length to the side of the cube.  I think 1 is a pretty good number.     

You'll note that every edge of the tetrahedron is the diagonal of a side of the cube.     

If 1 is the length of the cube side, then sqrt(2) is the length of the tetrahedron edge.     

The volume of a regular tetrahedron in terms of the length of its edge is given as:     

     V  =  a³ / 6 • sqrt(2)             where a is the length of the edge of the tetrahedron     

     V  =  sqrt(2)³ / 6 • sqrt(2)     divide that sqrt(2) in the denominator into the numerator     

     V  =  sqrt(2)² / 6     

     V  =  2 / 6  =  1 / 3     

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A kite is flying 82 ft. off the ground, and its string is pulled taut. The angle of elevation of the kite is 52°. Find the length of the string. Round your answer to the nearest tenth.     

     The string would be the hypotenuse of a right triangle.  sine = opposite / hypotenuse     

     sine(52)  =  82 / string       Use the trig tables or your calculator to find the sine of 52^o     

     0.7880  =  82 / string     

     string  =  82 / 0.7880     

     string  =  104.06  to nearest tenth is  104.1 ft     

_.     

     x = 26^o    

     You don't need all those calculations.  There's a theorem.     

     Theorem:  When a transversal line intersects two parallel lines,     

                       the alternate interior angles are equal in measure.     

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