Bosco

Among all rectangles that have a perimeter of 84, find the dimensions of the one whose area is largest. Write your answers as fractions reduced to lowest terms.     

The rectangle that has the highest ratio of area to perimeter is a square.     

Therefore, when its perimeter is 84, the sides of that square are 21x21.     

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Wat number can 6 and 9 go into  equally?     

Not counting zero, the first one is 18 . . . but there are infinitely more of them.     

_.     

  Nine points are arranged in a 3 x 3 grid, as shown below.     

  In how many different ways can you choose three points, so that they form a triangle?     

  72     

  Let me tell you how I got that number.     

  I think that any combination of three dots in that 3x3 grid will form a triangle, except     

  the eight triples that lie in a straight line; namely, 3 vertical, 3 horizontal, 2 diagonal.     

  So I went ₉C₃ = 84 and then subtracted the eight straight lines that would be formed.     

  I posted this answer a while ago, but made a typo.  I said 9x9 grid.  Should been 3x3.     

_.     

  Nine points are arranged in a 3 x 3 grid, as shown below.     

  In how many different ways can you choose three points, so that they form a triangle?    

  72     

  Let me tell you how I got that number. 

  I think that any combination of three dots in that 9x9 grid will form a triangle, except     

  the eight triples that lie in a straight line, namely, 3 vertical, 3 horizontal, 2 diagonal.     

  So I went ₉C₃ = 84 and then subtracted the eight straight lines that would be formed.     

_.     

Paul has a mix of quarters and dimes in his piggybank. He has twice as many dimes than he has quarters. He counts all his coins and notices that he has   nine dollars in total. How many quarters does Paul have?     

D  =  2Q     

10D + 25Q  =  900     

10(2Q) + 25Q  =  900     

   20Q   + 25Q  =  900     

                45Q  =  900     

                    Q  =  20     

                     D  =  2Q     

                     D  =  2 • 20     

                     D  =  40     

check answer     

20 quarters is 5 dollars     

40 dimes     is 4 dollars     

total is             9 dollars     

_.     

  In right triangle BCD with D = 90, we have BC = 9 and BD = 5. Find sin B.     

  As I always tell everybody, sketch the triangle.     

  sin(B) = BD / CD     

  We know what BC and BD are, so one way is to use Pythagoras' to find CD.     

  CD²  =  BC² – BD²     

  CD²  =  9² – 5²  =  81 – 25  =  56     

  CD    =  sqrt(56)  =  7.4833              ––>>   sin(B)  =  5 / 7.4833     

                                                                      sin(B)  =  0.6682     

_.     

In triangle ABC , we have A = 90 and sin B = 4/17. Find cos C.     

sin(B) = AC / BC    Well, whaddya know.  Sketch the triangle 

                               and you find that cos(C) also = AC / BC     

                                therefore            cos(C) = 4 / 17     

_.     

The value of the numerator of a particular fraction is one-half the value of its denominator. If the numerator is increased by 2 and the denominator is decreased by 2 , the value of the resulting fraction is 5/4 . What is the denominator of the original fraction?     

original fraction is   x / 2x     

therefore     

                                  x + 2           5     

                                ––––––   =   –––     

                                 2x – 2           4     

crossmultiply           (4)(x + 2)  =  (5)(2x – 2)     

                                     4x + 8  =  10x – 10     

                                        – 6x  =  – 18     

                                             x  =  3     

denominator of     

original fraction                   2x  =  6     

check answer     

(3 + 2) / (6 – 2)  =   5 / 4    True     

_.     

Circle O and circle P, with radii 3 and 8, respectively, are both tangent to line l at H. Enter all possible lengths of OP separated by commas.     

If they're on the same side of the line, OP = 5.     

If they're on opposite sides of the line, OP = 11.     

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