Lesson Classic entertainment problems

Classic entertainment problems

Problem 1

Problem 2

    A striking clock (also known as chiming clock) is a clock that sounds the hours audibly on a bell or gong. 
    In 12-hour striking, used most commonly in striking clocks today, the clock strikes once at one a.m., twice at two a.m., 
    continuing in this way up to twelve times at 12 noon, then starts again, striking once at one p.m., 
    twice at two p.m., up to twelve times at 12 midnight.

Problem 3

Let x be the width in meters of the roll of newspaper.

We need to use consistent units; so call the thickness of the newspaper 0.00022 m.


Spread out into a rectangular sheet, the newspaper is a very thin rectangular solid with length 7000 m, width x m, and thickness 0.00022 m.  

The volume is then 1.54x cubic meters.


When rolled into a cylinder, the volume will be the same.

The volume of a cylinder is (pi)*(radius squared)*(height); the height of the cylinder is the width of the newspaper.  So

 = 

 =  =    (approximately... 0.490197....)

 =  = 

The radius of the roll of newspaper is approximately 0.7 m.

Problem 4

According to the condition, the entire/(the whole) cake is cut into small (1 cm X 1 cm x 1 cm) cubes.


The total number of these small cubes is  9*14*15 = 1890.


Of them, have chocolate on at least one side those and only those cubes that are in one slice adjacent to some (to any) face of the prism.


The rest of the small cubes HAVE NO chocolate on any small faces.


Those small cubes that have no chocolate on any small face are ALL INTERNAL cubes.
They form smaller prism whose dimensions are 2 cm less than dimensions of the whole prism. 
Namely, these dimensions are (9-2) = 7 cm, (14-2) = 12 cm  and (15-2) = 13 cm.

So, the number of small INTERNAL cubes is 7*12*13 = 1092.


Thus the number of cubes that have chocolate at least on one side is  1890 - 1092 = 798.


The number of small cubes that have chocolate exactly on one side is

2*(7*12 + 7*13 + 12*13) = 662.


Answer.    The number of small cubes that have chocolate at least on one side is  798.

           The number of small cubes that have chocolate exactly on one side is 662.

Problem 5

Let b be the number of boys and g be the number of girls.

Then

b - 1 = g       (1)   ("each boy has as many sisters as brothers")
2*(g-1) = b     (2)   ("each girl has only half as many sisters as brothers")


First equation says "each boy has b-1 brothers, and this number is equal to the number of sisters".


Second equation says "each girl has g-1 sisters, and .. . and so on . . . ")


From (1), substitute "b" into (2) and get the answer.