Bosco

   simplify 4 to the power of 9 times 4 to the power of 3     

   4⁹ = (4 • 4 • 4 • 4 • 4 • 4 • 4 • 4 • 4)     

   4³ = (4 • 4 • 4)     

   4⁹ • 4³  =  (4 • 4 • 4 • 4 • 4 • 4 • 4 • 4 • 4) • (4 • 4 • 4)     

   4⁹ • 4³  =  4¹²     

   _.     

   Let a_1, a_2, a_3, \dots, a_8, a_9, a_{10} be an arithmetic sequence.     

   If $a_1 + a_3 = 5$ and $a_2 + a_4 = 6$, then find $a_1$.     

   a₁                                         =  a₁     

   a₂                                         =  a₁ + d     

   a₃ = a₂ + d = (a₁ + d) + d    =  a₁ + 2d     

   a₄ = a₃ + d = (a₁ + 2d) + d  =  a₁ + 3d     etc.     

   a₂ + a₄  =  (a₁ + d) + (a₁ + 3d)  =  2a₁ + 4d  =  6     

   a₁ + a₃  =  (a₁)       + (a₁ + 2d)  =  2a₁ + 2d  =  5     

   by subtraction                                           2d  =  1     

                                                                       d  =  0.5     

   2a₁ + 4d          =  6     

   2a₁ + (4)(0.5)  =  6     

                   2a₁  =  6 – 2  =  4     

                     a₁  =  2   

   .     

   Can someone please find the pattern in this sequence of numbers?     

   3,4,3     

   4,5,8     

   6,11,12     

   ?,10,14     

   If you consider the center number of each group a null, inserted only to make deducing     

   a pattern more difficult, then add together the outside numbers of each group, you get     

   3  +  3  =     6     

   4  +  8  =   12     

   6  + 12  =  18     Continue the arithmetic sequence of the sums, common difference 6.     

   ?  + 14  =  24     

   so     ?  =  24 – 14     

            ?  =  10     

   _.     

   Find the largest positive integer $n$ such that $n^3$ divides 15.     

   Every number will divide 15.     

   But if you mean n³ that evenly divides 15, then n = 1 is the only one.     

   _.     

Will and Grace are canoeing on a lake.  Will rows at $50$ meters per minute and Grace rows at $30$ meters per minute. Will starts rowing at $2$ p.m. from the west end of the lake, and Grace starts rowing from the east end of the lake at $2{:}45$ p.m. If they always row directly towards each other, and the lake is $2800$ meters across from the west side of the lake to the east side, at what time will the two meet?     

Will will row for x number of minutes     

Grace will row for (x – 45) minutes     

[ (50 m/min) • x min ]  +  [(30 m/min) • (x – 45 min)]  =  2800 m     

50x + 30x – 1350  =  2800     

                      80x  =  4150     

                          x  =  51.875     note that 0.875 min = 52.5 sec     

So the time they meet is 2:51:52 pm     

   I have to paint one side of a wall.  The wall is $15$ meters tall and $60$ meters long.     

   Each gallon of paint covers $400$ square feet. If a foot is approximately $0.3048$ meters, then     

   what is the smallest whole number of gallons I can buy and have enough paint to cover the whole wall?     

   1 ft = 0.3048 m     therefore     1 m = (1 / 0.3048) ft = 3.2808 ft     

   H _wall = (15)(3.2808) ft =   49.212 ft     

   L _wall  = (60)(3.2808) ft = 196.848 ft     and so     Area wall = (49.212 ft) x (196.848 ft)     

                                                                                                = 9687.284 sqft     

   Paint req'd     ––>>     (9687.284 sqft) / (400 sqft/gallon)  =  24.218 gal     

   You can't buy just 0.218 of a gallon, so you will need to buy 25 gallons.     

   _.     

   A quadrilateral has angles of $q$, $4q + 17^\circ $, $8q + 21^\circ$, and $15q + 148^\circ $.     

   Find the largest angle of the quadrilateral, in degrees.     

   q + (4q + 17) + (8q + 21) + (15q + 148)  =  360     

                                                 28q + 186  =  360     

                                                 28q            =  174     

                                                               q  =  6.214     

       q            =    6.214     

     4q +   17  =  41.856     

     8q +   21  =  70.712     

   15q + 148  =  241.210     

   _.     

   What is the square root of 30^2-16^2??     

   sqrt (30² – 16² )     

   sqrt (900 – 256)     

   sqrt (644)  =  25.3772  (rounded)     

   _.     

   Which weighs more a pound of sand or a pound of feathers?     

   LOL they weigh the same.     

   Here's a little more complex trick question for you:     

   Which weighs more, a pound of gold or a pound of feathers?     

   The feathers do, because gold is weighed using what they call the Troy system.     

   A Troy pound, weighed on your bathroom scale would register at 0.823 pounds.     

   That's because almost everything in the U.S. except precious metal is weighed     

   using what they call the avoirdupois system.  A Troy ounce weighs a little more     

   than an avoirdupois ounce, but there are only thirteen of them in a Troy pound.     

   _.     

   Pearl writes down seven consecutive integers, and adds them up.     

   The sum of the integers is equal to 28/5 times the largest of the seven integers.     

   What is the smallest integer that Pearl wrote down?     

   x + (x+1) + (x+2) + (x+3) + (x+4) + (x+5) + (x+6)  =  28/5 • (x+6)     

                                                                    7x + 21  =  (28x + 168) / 5     

                                                                35x + 105  =  28x + 168     

                                                                            7x  =  63     

                                                                              x  =  9     

   check answer     

   Does 9 + 10 + 11 + 12 + 13 + 14 + 15  =  28/5 • 15   ?     

                                                           84  =  84   True     

   _.