answers of week no. 27 i.e from 5 - 11 june 2010

1) .

The bear was white because it was a polar bear. The only place on earth where a bear can go south, west and north equal distances and end up where it started is the North Pole.

Actually, the bear could go west two or five kilometers instead of one and it would not make any difference -- the bear would be making a circle around the North Pole. East and West you travel along parallels which are circles equidistant from the poles. North and South you travel along meridians which are circles that cross both the north and the south poles.

There is another way to go south, west and north one kilometer and end up at the original point without starting at the North Pole^*.   Start 1 + 1/(2π) kilometers (approx. 1.16Km) from the South Pole. Walk one kilometer south, then one kilometer west (which is a complete 1 Km circle around the South Pole), and finally, one kilometer north to the starting point. However, there are no bears in Antarctica; there are only penguins.

At least one polar bear has actually been tracked by satellite traveling from Alaska to Greenland across the north pole region. Bears of other colors, e.g., grizzlies, Alaskan black bears, Russian brown bears, etc., never venture north of the continental land masses. Polar bears are not restricted by the boundaries of the land because they have partially webbed forepaws and are excellent swimmers. Their scientific name, Ursus maritimus, means "sea bear".

2).  

This is a classic problem that many people get wrong because they reason that half of a hen cannot lay an egg, and a hen cannot lay half an egg. However, we can get a satisfactory solution by treating this as a purely mathematical problem where the numbers represent averages.

To solve the problem, we first need to find the rate at which the hens lay eggs. The problem can be represented by the following equation, where RATE is the number of eggs produced per hen·day:

1-1/2 hens × 1-1/2 days × RATE = 1-1/2 eggs

We convert this to fractions thus:
3/2 hens × 3/2 days × RATE = 3/2 eggs

Multiplying both sides of the equation by 2/3, we get:
1 hen × 3/2 days × RATE = 1 egg

Multiplying both sides of the equation again by 2/3 and solving for RATE, we get:
RATE = 2/3 eggs per hen·day

Now that we know the rate at which hens lay eggs, we can calculate how many hens (H) can produce 12 eggs in six days using the following equation:

H hens × 6 days × 2/3 eggs per hen·day = 12 eggs

Solving for H, we get:
H = 12 eggs /(6 days × 2/3 eggs per hen·day) = 12/4 = 3 hens

Therefore, the farmer needs 3 hens to produce 12 eggs in 6 days.

3).  

This is a typical algebra problem with two unknowns, and we need to have two equations to solve the problem. Let X be the number of cows, and Y the number of chickens.

The number of cows plus the number of chickens equals 30:

X + Y = 30

Since cows have four legs and chickens have two legs, we can also have an equation for the number of legs.

4 X + 2 Y = 74

Rearranging the first equation we get

X = 30 - Y

Substituting X in the second equation gives us

4 × (30 - Y) + 2 Y = 74

120 - 4 Y + 2 Y = 74

-2 Y = 74 - 120 = -46

Y = 23

The farmer has 23 chickens. (and 7 cows).

4).  

The container was half full at 12:59 PM. When the bacteria doubled in the next minute, the container became full. This is an example of exponential growth where the growth rate is a mathematical function that is proportional to the function's current value.

5). 

The number of bacteria in the container is 2ⁿ where n is the number of minutes elapsed. The full container has 2⁶⁰ (two to the sixtieth power), or approximately 1.153×10¹⁸ bacteria. Since one micron (one millionth of a meter) is 1×10^-4 centimeters, each bacterium occupies a volume of about 1×10^-12 cubic centimeters. Multiplying times the number of organisms, we conclude that the container had a capacity of 1.153×10⁶ cc or 1,153 liters. The container was slightly bigger than one cubic meter.

      2⁶⁰ = 1,152,921,504,606,846,976