Question

A rabbit eats a certain number of carrots placed along a path of distance 10m. the rabbits eats 1 carrot after covering a distance of 1m. after covering every meter, he eats double the number of carrots. how much distance does the rabbit cover if it eats exactly half the number of carrots placed along the path

Answer 1

Answer:    

we know from the first example that a rabbit takes 3 mins to eat a carrot. If all carrots were of an exactly equal size, each rabbit (however many there are) will take 9 minutes to eat 3 carrots. In your example 6 rabbits x 3 carrots each is 18 carrots.

Step-by-step explanation:

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Answer 2

The distance required to cover to eat half of the carrots according to the question requirement is 9m.

Given :

Length of the path = 10m

Number of carrots at 1m = 1

Number of carrots doubles with every meter

To Find :

The distance to cover to eat half of the total number of carrots

Solution :

Initially in the path i.e. at 1 km distance the number of carrots = 1

At 1 km, the Number of carrots = 1 i.e. $2^{0}$

At 2 km, the Number of carrots = 1*2 = 2 i.e. $2^{1}$

At 3 km, the Number of carrots = 2*2 = 4 i.e. $2^{2}$

At 4 km, the Number of carrots = 2*4 = 8 i.e. $2^{3}$

Thus in general, at a distance of x km, the number of carrots are $2^{x-1}$.

Total number of carrots placed along the 10 km path

              = Carrots at 1km + Carrots at 2km + ..... + Carrots at 10km

              $= 2^{0} + 2^{1}+2^{2}+2^{3}.....+2^{9}$

              $= 2^{10} - 1$.    [ as $2^{0}+2^{1}+...+2^{n-1} = 2^{n} -1]$

              $= 1024 - 1$

              $= 1023$

Half the number of total carrots placed along the path $= 1023 / 2$

                                                                                           $= 511 ( approx )$

Let x km be the distance the rabbit needs to cover to eat half of the total number of carrots placed.

Total carrots eaten at x km distance $= 2^{0} + 2^{1} + 2^{2} +.....+ 2^{x-1}$

                                                             $= 2^{x} - 1$

As this is equal to half the total number of carrots,

                                         $2^{x} -1 = 511$

                                         $2^{x} = 512$

                                         $2^{x} = 2^{9}$

                                          $x = 9$

Thus after covering a distance of 9 km, the number of carrots eaten by the rabbit is half of the total carrots placed along the path.

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