It Only Takes A Flower

The lake would be full when each flower occupies half of the lake.

Since the flower doubles each day & covers the lake in $30$ days, it will cover half the lake in $29$ days.

$\Rightarrow$ Both the flowers would cover the lake in $\boxed{29}$ days.

$Let,initial\quad size\quad of\quad flower=x\quad \& \quad size\quad of\quad lake=y.\\ Flower\quad takes\quad 'n'\quad days\quad to\quad occupy\quad the\quad entire\quad lake.\\ \\ For\quad 1^{ st }\quad day,\quad size\quad of\quad the\quad flower=x\\ for\quad 2^{ nd }\quad day,\quad size\quad of\quad the\quad flower=2x\\ for\quad 3^{ rd }\quad day,\quad size\quad of\quad the\quad flower=4x\\ for\quad 4^{ th }\quad day,\quad size\quad of\quad the\quad flower=8x\\ so,\quad n^{ th }\quad day,\quad size\quad of\quad the\quad flower={ 2 }^{ n-1 }x,\quad which\quad is\quad the\quad size\quad of\quad lake.\\ So, y={ 2 }^{ n-1 }\times x\\ We\quad know\quad it\quad takes\quad 30\quad days.\quad So,\quad y={ 2 }^{ 30-1 }x={ 2 }^{ 29 }x.\quad \\ If\quad there\quad are\quad two\quad flowers\quad in\quad same\quad size\quad \& \quad they\quad takes\quad 'm'\quad days\quad to\quad occupy\quad the\quad lake.\\ ({ 2 }^{ m-1 }x)+({ 2 }^{ m-1 }x)={ 2 }^{ 29 }x\\ \Rightarrow 2.{ 2 }^{ m-1 }x={ 2 }^{ 29 }x\\ \Rightarrow { 2 }^{ m }={ 2 }^{ 29 }\\ \Rightarrow m=29\\ So,\quad they\quad takes\quad 29\quad days\quad to\quad occupy\quad the\quad lake.\\$

For one flower 1 2 4 8 16.... Till 2^30 Are the flowers on 1 2 3 4 5 days respectively

For two flowers it will be 2 4 8 16 .... Till 2^30 This has just one missing term i.e. 1

Hence answer is 30 -1 days = 29

• First of all, you start with one single flower, and each day the size of that flower multiplies by a factor of 2.

So, the size of the pond is $1 \times 2^{30}$ times the size of the flower. That equals to $1024 \times 1024 \times 1024$ .

• Second of all, another flower appeared, and the equation became :

$2 \times 2^{n} = 1024 \times 1024 \times 1024$

$2^{n} = 512 \times 1024 \times 1024$

$2^{n} = 2^{9} \times 2^{10} \times 2^{10}$

$2^{n} = 2^{29}$

$n = \boxed{29}$

It takes 30 days to fill the entire lake, and since it doubles the size everyday, on the 29h day it would fill half of the lake, and 2 of such flowers combine will fill the entire lake.

Since the flower occupies the whole lake on the 30th day and it doubles in size everyday, the flower was half the size of the lake on the 29th day. So 2 flowers would occupy the whole lake.

Situation 1) Let x be the initial size of the flower and y be the size of lake. We have got $\boxed {x+ 2x+ 4x+...+ (2^{29}) x = y}$ With the help of GP summation formula we get $\boxed{y = x(2^{29} - 1)}$ .

Situation 2) Now we have got two such flowers. Therefore, $\boxed{2[ x+ 2x+ 4x+ ...+ (2^n)x ] = y }$ (where n is the number of days to occupy the entire lake in stituation 2) Substituting value of y from situation 1 and solving we get $\boxed{n=29}$

Since the size of the flower gets doubled every day, it occupies exactly half of the lake at the end of 29th day.If there are two such flowers,on the 29th day itself,the lake gets completely occupied.

Given us a flower doubles for every day. for the first day : 2^1 for the second day : 2^2 for the third day : 2^3 and so on... for nth day : 2^n. Given it takes 30 days to occupy the entire lake for one flower then it will take 29 days to occupy the half of lake. Since we have two flowers it will take 29 days to occupy entire lake.

Let F be the area of such flower. F(2)^x be the expression that the area of F double every x day(s). If F doubles for 30 days then, F(2)^30. If there's two F's then, (2)(F)2^30, but that exceeds the maximum area of the lake covered by one flower in 30 days. Therefore, (2)(F)2^(30-1). Thus, x should equal 29 if there are 2 flowers of the kind

This problem is very simple and can be solved without a pen and paper as other people have previously shown.
But it is always good to test our knowledge on how to show things in a more formal way.
Solving it in paper can be done like this:

$2^{30}x = y$
$2^{a}x + 2^{a}x = y$
$2(2^{a}x) = 2^{30}x$
$2^{a}x = 2^{29}x$
$a = 29$

If there are now 2 flowers sharing the lake, you just need to divide the lake in half for them.

Since the flowers are always multiplying themselves by 2 every day though, sharing their container only takes away one of their days.

In the 29th day, one flower occupies half the lake. Therefore if we had two flowers, that additional flower could've filled the other half independently. Therefore 29 days must be the answer.

Supose Flower size is 2 inch.

In 30 days, it will fill x dimension of the lake

Such as: 2^30 = x

Then, 2 flowers have a dimension of 4inch

So: 4^y = x

But its the same that: (2*2)^y = x

And we know, 2^30 = x

So: (2*2)^y = 2^30 And then: 2(2^y) = 2^30

Therefore, y = 29

Doubt: Im not sure why it works with size=2 and any other, on this method, if someone could please help me...

If the flower takes 30 days to cover entire lake of area A and doubles every day, then the area of flower must be A/2^30. Now area of two flowers will be 2A/2^30 , which is equal to A/2^29 . So it takes 29 days to cover entire lake.

While I love the intuition behind the solution of this problem, the way I solved this was:

First, I noticed that the size of flower on the $30^{th}$ day is $2^{29}x$ , when $x$ is the size it all started with.

It's easy to notice, that we could count the size of any flower with such formula, when $y$ is days:

$2^{y-1}x$

Now, we can make a basic equation:

$2^{29}x=2^{y-1}x + 2^{y-1}x$

Which simplifies to:

$2^{29}x=2^{y}x$

$x = 29$

That's the solution for pure math lovers ;)

1 flower = x

1 flower in the lake = x • 2^numbers of days

n = number of days

1 flower in the lake: x • 2^n

Lake = x • 2 • 2 • 2 • 2....

Lake = x • 20^30

2 flowers:

x • 2^n + x • 2^n = 2 • 2^n

2 • 2^n = 2^n+1

2 flowers = x • 2^n+1

Lake = x • 2^30

Lake = 2 flowers

x • 2^30 = x • 2^n+1

2^30 = 2^n+1

n+1 = 30

n = 29

Well, the flower doubles every day, so if you are starting with 2 flowers, think of the original 30 but instead of starting on Day 1, you are starting on Day 2, so one less, 29

It doubles everyday, so starting with two just subtracts one day

In 29 days, one fill half the lake. Similarly the other flower also fill half the lake in these days. So the lake in fullin 29 days.

Let the volume of the lake be V and the volume of the flower after day 1 of its birth be p (necessary assumption: V > 0 => p > 0). => V = (2^29)p = (2^n)p + (2^n)p = [2^(n+1)]p where n is the no. of days it takes for the two flowers to fill the lake decremented by 1 (n is real). => 2^29 = 2^(n+1) => n = 28 => It takes 29 days for the two flowers to fill the lake. [Q.E.D.]

If the size double every day and cover the lake in 30 days. On day 29 this flower will cover half the lake so it will take another to cover the the other half.

For the first flower to fill up the lake, you would take the flower's size F times 2^30 because it doubles itself 30 times. Then, you divide that by F x2 to find how much you would multiply 2 flowers by to fill the lake, giving you 2^(30-1) or 2^29. Therefore, it would take the two flowers 29 days to fill the lake.

Used a different method. Say the flower's size is x=2 on day 1. On day 30, the size of the flower will be 2^30 which is equal to 1073741824 which covers the entire lake so therfore the size of the lake is also 1073741824. For one flower to cover half the lake, it has to be half the size which is 536870912 and the second half will also be the same. It'll take 29 days for the flower to grow to that number. 2^29 is 536870912.

Lets suppose the first day the flower was kept in the lake it occupied 1 unit of area. Then on the second day, it shall occupy 2 units of area, 4 units on third and so on. That is on a nth day, the total area occupied by the flower is (2^ n -1) ----(1)

Since the flower takes total of 30 days to occupy the lake, area of the flower on the 30th day is the area of lake. By (1) it is 2 ^ 29 units.

Now the total area occupied by two flowers on any given day n is 2 * ( area occupied by one flower on any given day n) = 2* 2^ n-1 OR 2^n units.

The day the lake gets full is when the total area occupied by both the flowers is equal to the area of the lake. Since the lake area is 2^29, hence 2 ^ n = 2^29 OR n = 29 days.

Let size of one flower= x At the end of 30 days, size of flower= x* 2^30 Size of two flowers= 2x Required time=( x*2^30)/2x

Question is imprecise because it says "how long" not "how many days". The correct answer should be "29 days" but "696 hours" would also be correct.

Since it takes 30 days to cover the entire lake , lake is half filled at 29th day.so, both flowers takes 29days to cover the entire lake..

may be geometric progression can help understand those who do not understand

Took longer than I care to admit to simply think backwards and see how much of the lake would be occupied on the 29th day...

I did way too much work instead of thinking it out logically. I had 2(2^x)=1073741824 log (base 2) 536870912 = x ln 536870912 / ln 2 29 Like I said, thinking logically would have saved a lot of time.

29 is the answer because.. 1 flower takes 30days to occuppy the entire lake, meaning in the 29th day the lake is half occuppied by a 1 flower,, so if you add another 1 flower in the lake it takes the other half of the lake in the 29th day...

Covering lake while getting doubled in size for 30 days is:

2^30.

For two flowers, width parameter will be halfed,

2^30/2

2^29

So, both floweres will be doubled for 29days to fill the pound

We can say that 2^30 = 2(2^n), since there is two flowers We then factor out a 2, making it 2(2^29) = 2(2^n) therefore n=29

The problem can be solved by Geometric Progression wherein the common ratio r = 2, n will be defined as the number of days, every number in the sequence represents the factor the area is increase for every day. Let A = area covered by a single flower in day 1 Since in GP, the nth term can be solve using $a_n=a_1\times r^{n-1}$ For a single flower,

day 1, $a_1=A\times 2^0=A$

day 2, $a_2=A\times 2^1=2A$ means the area is doubled,

day 3, $a_3=A\times 2^2=4A$ double of day 2,

day 4, $a_4=A\times 2^3=8A$ double of day 3,

until the 30th day, $a_{30}=A\times 2^{29}$ , this also represents the area of the lake to be filled up by the 2 flowers.

Now for 2 flowers,

day 1, $a_1=2A\times 2^0=2A$

day 2, $a_2=2A\times 2^1=4A$

day 3, $a_3=2A\times 2^2=8A$

until such day (the unknown) where $a_n=A\times 2^{29}$ , substitute this for the work done of 2 flowers,

$a_n=2A\times 2^{n-1}$

$A\times 2^{29}=2A\times 2^{n-1}$ the parameter area A can be cancelled,

$2^{29}=2\times 2^{n-1}$

$\frac{2^{29}}{2}=2^{n-1}$

$2^{28}=2^{n-1}$ and since we have the same base on both sides, we can simplify it... to solve for n

$28=n-1$

$n=29$ days for two flowers to occupy the entire lake.

Long method:Using the formulas of a geometric progression Tn=ar^(n-1)

With 1 flower in the pond, the progression is as the following:

1,2,4,8,16

r=2

Therefore, in 30 days, Tn = ar^(n-1) T30 = 12^(29) = 536 870 912 flowers in the pond

With 2 flowers, the progression is as the following:

2,4,8,16,32

r = 2

In "n" number of days, the pond will be full (536 870 912 flowers) .Therefore:

Tn = ar^(n-1) = 536 870 912

Tn = 2(2)^(n-1) = 536 870 912

Tn = 2(2)^(n-1) = 2^29

Tn = 2^(n-1) = 2^28

n - 1 = 28

n = 29 days.

I have read all your solutions, and here is mine: If we got 2 flowers, after 30 days, they will occupy 2 lakes, so the previous day, they occupied the whole lake The answer is 29 days!

Hello,

as for a flower to occupy a single lake is 30 days ,where n = 30, it is 2^n

numbers of flowers in 30 days = 2^30,

therefore,number of days to fill the lake for( 2 flowers) = 2^30 / 2 = 2^29 as 2^n,

2^n = 2^29

n = 29 days...

thanks...

$2\cdot 2^{n-1}=2^{30-1}$

$2^{n-1+1}=2^{29}$

$2^n=2^{29}$

$\therefore n=29$

The lake would be full in 30 days with one flower....so by 29th day it will occupy half the space..so if there are 2 flowers then by 29th day the lake will be fully occupied..

Let x be the initial size of the flower. Let the size of lake be y. The follows the sequence x, 2x, 4x..., 2^30x. Now, 2^30x = y. With 2 flowers let the time taken be n days. 2^n * x + 2^n * x = y. Which gives 2^(n+1) * x = y. But we already know that 2^30 = y. n=29. Hence answer is 29 days.

29th day 1 flower will cover the half lake... therefore remainig lake would also be covered within same time by another flower...

Since the flowers get doubled every day,lake will be half covered on 29th day and full covered on 30th day,so both flowers will get covered the lake in 29 days.

K.K..GARG India

One flower takes 30 days to grow.

Therefore, initial size of the flower =2^-30

Initial size of two flowers =2^-30 *2 = 2^-29

Since it doubles everyday, you require 29 days

in the first case one flower will get double to be 2 flowers on one day. the second case start from the day one so it will cover the lake in ( 30day - 1day =29day)