One lily pad, doubling in size every day, covers a pond in 30 days. How long would it take eight lily pads to...












6















A lily pad sits on a pond. It doubles in size every day. It takes 30
days for it to cover the pond. If you start with 8 lily pads instead,
how many days does it take to cover the pond?




I think that the answer is $27$, but I don't really think that makes sense intuitively. I think that, intuitively, the answer should be less than $30/4$ since it is increasing at an exponential rate.










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  • 1




    Hint: Set up an algebraic equation relating to what the word problem is saying. You should see your answer then fairly easily.
    – John Omielan
    22 hours ago






  • 3




    It'll take $27$ days only. Think of it like this that (for the first case) on the second day no of lily pads will become $2$, on the third day $4$ and on the fourth $8$. In your second case you start your first day at this point
    – Sauhard Sharma
    22 hours ago








  • 6




    Just out of curiosity, where does 30/4 come from Joseph?
    – Peter
    18 hours ago


















6















A lily pad sits on a pond. It doubles in size every day. It takes 30
days for it to cover the pond. If you start with 8 lily pads instead,
how many days does it take to cover the pond?




I think that the answer is $27$, but I don't really think that makes sense intuitively. I think that, intuitively, the answer should be less than $30/4$ since it is increasing at an exponential rate.










share|cite|improve this question




















  • 1




    Hint: Set up an algebraic equation relating to what the word problem is saying. You should see your answer then fairly easily.
    – John Omielan
    22 hours ago






  • 3




    It'll take $27$ days only. Think of it like this that (for the first case) on the second day no of lily pads will become $2$, on the third day $4$ and on the fourth $8$. In your second case you start your first day at this point
    – Sauhard Sharma
    22 hours ago








  • 6




    Just out of curiosity, where does 30/4 come from Joseph?
    – Peter
    18 hours ago
















6












6








6


1






A lily pad sits on a pond. It doubles in size every day. It takes 30
days for it to cover the pond. If you start with 8 lily pads instead,
how many days does it take to cover the pond?




I think that the answer is $27$, but I don't really think that makes sense intuitively. I think that, intuitively, the answer should be less than $30/4$ since it is increasing at an exponential rate.










share|cite|improve this question
















A lily pad sits on a pond. It doubles in size every day. It takes 30
days for it to cover the pond. If you start with 8 lily pads instead,
how many days does it take to cover the pond?




I think that the answer is $27$, but I don't really think that makes sense intuitively. I think that, intuitively, the answer should be less than $30/4$ since it is increasing at an exponential rate.







sequences-and-series algebra-precalculus






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edited 12 mins ago









Blue

47.7k870151




47.7k870151










asked 22 hours ago









josephjoseph

46710




46710








  • 1




    Hint: Set up an algebraic equation relating to what the word problem is saying. You should see your answer then fairly easily.
    – John Omielan
    22 hours ago






  • 3




    It'll take $27$ days only. Think of it like this that (for the first case) on the second day no of lily pads will become $2$, on the third day $4$ and on the fourth $8$. In your second case you start your first day at this point
    – Sauhard Sharma
    22 hours ago








  • 6




    Just out of curiosity, where does 30/4 come from Joseph?
    – Peter
    18 hours ago
















  • 1




    Hint: Set up an algebraic equation relating to what the word problem is saying. You should see your answer then fairly easily.
    – John Omielan
    22 hours ago






  • 3




    It'll take $27$ days only. Think of it like this that (for the first case) on the second day no of lily pads will become $2$, on the third day $4$ and on the fourth $8$. In your second case you start your first day at this point
    – Sauhard Sharma
    22 hours ago








  • 6




    Just out of curiosity, where does 30/4 come from Joseph?
    – Peter
    18 hours ago










1




1




Hint: Set up an algebraic equation relating to what the word problem is saying. You should see your answer then fairly easily.
– John Omielan
22 hours ago




Hint: Set up an algebraic equation relating to what the word problem is saying. You should see your answer then fairly easily.
– John Omielan
22 hours ago




3




3




It'll take $27$ days only. Think of it like this that (for the first case) on the second day no of lily pads will become $2$, on the third day $4$ and on the fourth $8$. In your second case you start your first day at this point
– Sauhard Sharma
22 hours ago






It'll take $27$ days only. Think of it like this that (for the first case) on the second day no of lily pads will become $2$, on the third day $4$ and on the fourth $8$. In your second case you start your first day at this point
– Sauhard Sharma
22 hours ago






6




6




Just out of curiosity, where does 30/4 come from Joseph?
– Peter
18 hours ago






Just out of curiosity, where does 30/4 come from Joseph?
– Peter
18 hours ago












5 Answers
5






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26














Starting with 8 pcs. is like 3 days have gone by. So 27 days remain.






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    8














    Hint $#1$:



    At the end of the $30$ days with one lilypad, the doubling means that the lilypad now encompasses the area of $2^{30}$ of the original lilypads.



    In that light, starting with $2^3 = 8$ lilypads and each one doubling in size per day, how many doublings will it take for you to get to $2^{30}$?



    (I know you've already solved this, I just think rewording/reframing the question might make it a bit easier to grasp on the intuitive level.)





    Hint $#2$:



    If that doesn't help ease your intuition behind your answer (which to my understanding is correct), keep in mind that starting with $8$ lilypads is basically no different than your first scenario after $3$ days. Sure, you have more lilypads, but since each doubles in size, it's no different than one lilypad of the same size as those $8$ put together then doubling. The number of lilypads differs, but we're focused on the total area encompassed.






    share|cite|improve this answer





























      6














      Your answer is correct



      If we denote the size of the lily pad by $x$, then after $1$ day, the coverage becomes $2x$ , . . . Also let's denote the size of pond by $y$. The assumptions imposes that:$$2^{30}xge y$$and $$2^{29}x<y$$Now starting with $8$ lily pads we obtain $$2^{27}cdot 8xge y$$and$$2^{26}cdot 8x< y$$which shows that the completion of the process takes long as much as $27$ days.



      The intuition is that:



      If we start with $1$ lily pad, after $3$ days we will have $8$ of them since the size of lily pads doubles each day. Starting with $8$ lily pads in first place and observing their growth is equivalent to observe the growth of $1$ lily pad during the days skipping the first $3$ days. If whole the process takes long $30$ days, then or modified process takes long $30-3=27$ days which is totaly intuitive.






      share|cite|improve this answer































        5














        $27$ is the right answer. Consider at which point the original lily pad is 8 times its original size (after three days), and how long it takes it to cover the lake from there.



        Also, this relative "indifference" to a seemingly large disparity in starting point ("$x$ times more to start with means it takes $30-y$ days rather than $30/y$") is exactly what exponential growth means.






        share|cite|improve this answer































          3















          I think that, intuitively, the answer should be less than $30/4$ since it is increasing at an exponential rate.




          Lily pad doubles in size every day, so it is increasing as a geometric progression.
          $$begin{array}{c|c|c|c|c|c|c|c|c}
          text{n-th day}&1&2&3&4&cdots&26&27&28&29&30\
          hline
          text{size of $1$ lily pad}&1&2&2^2&2^3&cdots&2^{25}&2^{26}&2^{27}&2^{28}&color{red}{2^{29}} end{array}$$



          If you start with $8$ lily pads, each doubling on its own, then:
          $$begin{array}{c|c|c|c|c|c|c|c|c}
          text{n-th day}&1&2&3&4&cdots&26&27&28&29&30\
          hline
          text{size of $8$ lily pads}&2^3&2^4&2^5&2^6&cdots&2^{28}&color{red}{2^{29}}&2^{30}&2^{31}&2^{32} end{array}$$

          Because when each of $8$ lily pads keeps doubling per day, the $8$ lily pads increase $8$ times faster in size altogether than that of one lily pad. So, you must multiply the size of one lily pad on any day by $8=2^3$ to find the total size of $8$ lily pads.



          As this source informs, the Giant Water Lily may grow as large as $8$ to $9$ feet ($2.4-2.7$m) in diameter.






          share|cite|improve this answer





















            Your Answer





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            5 Answers
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            5 Answers
            5






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            26














            Starting with 8 pcs. is like 3 days have gone by. So 27 days remain.






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              26














              Starting with 8 pcs. is like 3 days have gone by. So 27 days remain.






              share|cite|improve this answer
























                26












                26








                26






                Starting with 8 pcs. is like 3 days have gone by. So 27 days remain.






                share|cite|improve this answer












                Starting with 8 pcs. is like 3 days have gone by. So 27 days remain.







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                answered 21 hours ago









                zolizoli

                16.7k41845




                16.7k41845























                    8














                    Hint $#1$:



                    At the end of the $30$ days with one lilypad, the doubling means that the lilypad now encompasses the area of $2^{30}$ of the original lilypads.



                    In that light, starting with $2^3 = 8$ lilypads and each one doubling in size per day, how many doublings will it take for you to get to $2^{30}$?



                    (I know you've already solved this, I just think rewording/reframing the question might make it a bit easier to grasp on the intuitive level.)





                    Hint $#2$:



                    If that doesn't help ease your intuition behind your answer (which to my understanding is correct), keep in mind that starting with $8$ lilypads is basically no different than your first scenario after $3$ days. Sure, you have more lilypads, but since each doubles in size, it's no different than one lilypad of the same size as those $8$ put together then doubling. The number of lilypads differs, but we're focused on the total area encompassed.






                    share|cite|improve this answer


























                      8














                      Hint $#1$:



                      At the end of the $30$ days with one lilypad, the doubling means that the lilypad now encompasses the area of $2^{30}$ of the original lilypads.



                      In that light, starting with $2^3 = 8$ lilypads and each one doubling in size per day, how many doublings will it take for you to get to $2^{30}$?



                      (I know you've already solved this, I just think rewording/reframing the question might make it a bit easier to grasp on the intuitive level.)





                      Hint $#2$:



                      If that doesn't help ease your intuition behind your answer (which to my understanding is correct), keep in mind that starting with $8$ lilypads is basically no different than your first scenario after $3$ days. Sure, you have more lilypads, but since each doubles in size, it's no different than one lilypad of the same size as those $8$ put together then doubling. The number of lilypads differs, but we're focused on the total area encompassed.






                      share|cite|improve this answer
























                        8












                        8








                        8






                        Hint $#1$:



                        At the end of the $30$ days with one lilypad, the doubling means that the lilypad now encompasses the area of $2^{30}$ of the original lilypads.



                        In that light, starting with $2^3 = 8$ lilypads and each one doubling in size per day, how many doublings will it take for you to get to $2^{30}$?



                        (I know you've already solved this, I just think rewording/reframing the question might make it a bit easier to grasp on the intuitive level.)





                        Hint $#2$:



                        If that doesn't help ease your intuition behind your answer (which to my understanding is correct), keep in mind that starting with $8$ lilypads is basically no different than your first scenario after $3$ days. Sure, you have more lilypads, but since each doubles in size, it's no different than one lilypad of the same size as those $8$ put together then doubling. The number of lilypads differs, but we're focused on the total area encompassed.






                        share|cite|improve this answer












                        Hint $#1$:



                        At the end of the $30$ days with one lilypad, the doubling means that the lilypad now encompasses the area of $2^{30}$ of the original lilypads.



                        In that light, starting with $2^3 = 8$ lilypads and each one doubling in size per day, how many doublings will it take for you to get to $2^{30}$?



                        (I know you've already solved this, I just think rewording/reframing the question might make it a bit easier to grasp on the intuitive level.)





                        Hint $#2$:



                        If that doesn't help ease your intuition behind your answer (which to my understanding is correct), keep in mind that starting with $8$ lilypads is basically no different than your first scenario after $3$ days. Sure, you have more lilypads, but since each doubles in size, it's no different than one lilypad of the same size as those $8$ put together then doubling. The number of lilypads differs, but we're focused on the total area encompassed.







                        share|cite|improve this answer












                        share|cite|improve this answer



                        share|cite|improve this answer










                        answered 22 hours ago









                        Eevee TrainerEevee Trainer

                        5,2091834




                        5,2091834























                            6














                            Your answer is correct



                            If we denote the size of the lily pad by $x$, then after $1$ day, the coverage becomes $2x$ , . . . Also let's denote the size of pond by $y$. The assumptions imposes that:$$2^{30}xge y$$and $$2^{29}x<y$$Now starting with $8$ lily pads we obtain $$2^{27}cdot 8xge y$$and$$2^{26}cdot 8x< y$$which shows that the completion of the process takes long as much as $27$ days.



                            The intuition is that:



                            If we start with $1$ lily pad, after $3$ days we will have $8$ of them since the size of lily pads doubles each day. Starting with $8$ lily pads in first place and observing their growth is equivalent to observe the growth of $1$ lily pad during the days skipping the first $3$ days. If whole the process takes long $30$ days, then or modified process takes long $30-3=27$ days which is totaly intuitive.






                            share|cite|improve this answer




























                              6














                              Your answer is correct



                              If we denote the size of the lily pad by $x$, then after $1$ day, the coverage becomes $2x$ , . . . Also let's denote the size of pond by $y$. The assumptions imposes that:$$2^{30}xge y$$and $$2^{29}x<y$$Now starting with $8$ lily pads we obtain $$2^{27}cdot 8xge y$$and$$2^{26}cdot 8x< y$$which shows that the completion of the process takes long as much as $27$ days.



                              The intuition is that:



                              If we start with $1$ lily pad, after $3$ days we will have $8$ of them since the size of lily pads doubles each day. Starting with $8$ lily pads in first place and observing their growth is equivalent to observe the growth of $1$ lily pad during the days skipping the first $3$ days. If whole the process takes long $30$ days, then or modified process takes long $30-3=27$ days which is totaly intuitive.






                              share|cite|improve this answer


























                                6












                                6








                                6






                                Your answer is correct



                                If we denote the size of the lily pad by $x$, then after $1$ day, the coverage becomes $2x$ , . . . Also let's denote the size of pond by $y$. The assumptions imposes that:$$2^{30}xge y$$and $$2^{29}x<y$$Now starting with $8$ lily pads we obtain $$2^{27}cdot 8xge y$$and$$2^{26}cdot 8x< y$$which shows that the completion of the process takes long as much as $27$ days.



                                The intuition is that:



                                If we start with $1$ lily pad, after $3$ days we will have $8$ of them since the size of lily pads doubles each day. Starting with $8$ lily pads in first place and observing their growth is equivalent to observe the growth of $1$ lily pad during the days skipping the first $3$ days. If whole the process takes long $30$ days, then or modified process takes long $30-3=27$ days which is totaly intuitive.






                                share|cite|improve this answer














                                Your answer is correct



                                If we denote the size of the lily pad by $x$, then after $1$ day, the coverage becomes $2x$ , . . . Also let's denote the size of pond by $y$. The assumptions imposes that:$$2^{30}xge y$$and $$2^{29}x<y$$Now starting with $8$ lily pads we obtain $$2^{27}cdot 8xge y$$and$$2^{26}cdot 8x< y$$which shows that the completion of the process takes long as much as $27$ days.



                                The intuition is that:



                                If we start with $1$ lily pad, after $3$ days we will have $8$ of them since the size of lily pads doubles each day. Starting with $8$ lily pads in first place and observing their growth is equivalent to observe the growth of $1$ lily pad during the days skipping the first $3$ days. If whole the process takes long $30$ days, then or modified process takes long $30-3=27$ days which is totaly intuitive.







                                share|cite|improve this answer














                                share|cite|improve this answer



                                share|cite|improve this answer








                                edited 21 hours ago

























                                answered 21 hours ago









                                Mostafa AyazMostafa Ayaz

                                14.5k3937




                                14.5k3937























                                    5














                                    $27$ is the right answer. Consider at which point the original lily pad is 8 times its original size (after three days), and how long it takes it to cover the lake from there.



                                    Also, this relative "indifference" to a seemingly large disparity in starting point ("$x$ times more to start with means it takes $30-y$ days rather than $30/y$") is exactly what exponential growth means.






                                    share|cite|improve this answer




























                                      5














                                      $27$ is the right answer. Consider at which point the original lily pad is 8 times its original size (after three days), and how long it takes it to cover the lake from there.



                                      Also, this relative "indifference" to a seemingly large disparity in starting point ("$x$ times more to start with means it takes $30-y$ days rather than $30/y$") is exactly what exponential growth means.






                                      share|cite|improve this answer


























                                        5












                                        5








                                        5






                                        $27$ is the right answer. Consider at which point the original lily pad is 8 times its original size (after three days), and how long it takes it to cover the lake from there.



                                        Also, this relative "indifference" to a seemingly large disparity in starting point ("$x$ times more to start with means it takes $30-y$ days rather than $30/y$") is exactly what exponential growth means.






                                        share|cite|improve this answer














                                        $27$ is the right answer. Consider at which point the original lily pad is 8 times its original size (after three days), and how long it takes it to cover the lake from there.



                                        Also, this relative "indifference" to a seemingly large disparity in starting point ("$x$ times more to start with means it takes $30-y$ days rather than $30/y$") is exactly what exponential growth means.







                                        share|cite|improve this answer














                                        share|cite|improve this answer



                                        share|cite|improve this answer








                                        edited 21 hours ago

























                                        answered 22 hours ago









                                        ArthurArthur

                                        111k7107189




                                        111k7107189























                                            3















                                            I think that, intuitively, the answer should be less than $30/4$ since it is increasing at an exponential rate.




                                            Lily pad doubles in size every day, so it is increasing as a geometric progression.
                                            $$begin{array}{c|c|c|c|c|c|c|c|c}
                                            text{n-th day}&1&2&3&4&cdots&26&27&28&29&30\
                                            hline
                                            text{size of $1$ lily pad}&1&2&2^2&2^3&cdots&2^{25}&2^{26}&2^{27}&2^{28}&color{red}{2^{29}} end{array}$$



                                            If you start with $8$ lily pads, each doubling on its own, then:
                                            $$begin{array}{c|c|c|c|c|c|c|c|c}
                                            text{n-th day}&1&2&3&4&cdots&26&27&28&29&30\
                                            hline
                                            text{size of $8$ lily pads}&2^3&2^4&2^5&2^6&cdots&2^{28}&color{red}{2^{29}}&2^{30}&2^{31}&2^{32} end{array}$$

                                            Because when each of $8$ lily pads keeps doubling per day, the $8$ lily pads increase $8$ times faster in size altogether than that of one lily pad. So, you must multiply the size of one lily pad on any day by $8=2^3$ to find the total size of $8$ lily pads.



                                            As this source informs, the Giant Water Lily may grow as large as $8$ to $9$ feet ($2.4-2.7$m) in diameter.






                                            share|cite|improve this answer


























                                              3















                                              I think that, intuitively, the answer should be less than $30/4$ since it is increasing at an exponential rate.




                                              Lily pad doubles in size every day, so it is increasing as a geometric progression.
                                              $$begin{array}{c|c|c|c|c|c|c|c|c}
                                              text{n-th day}&1&2&3&4&cdots&26&27&28&29&30\
                                              hline
                                              text{size of $1$ lily pad}&1&2&2^2&2^3&cdots&2^{25}&2^{26}&2^{27}&2^{28}&color{red}{2^{29}} end{array}$$



                                              If you start with $8$ lily pads, each doubling on its own, then:
                                              $$begin{array}{c|c|c|c|c|c|c|c|c}
                                              text{n-th day}&1&2&3&4&cdots&26&27&28&29&30\
                                              hline
                                              text{size of $8$ lily pads}&2^3&2^4&2^5&2^6&cdots&2^{28}&color{red}{2^{29}}&2^{30}&2^{31}&2^{32} end{array}$$

                                              Because when each of $8$ lily pads keeps doubling per day, the $8$ lily pads increase $8$ times faster in size altogether than that of one lily pad. So, you must multiply the size of one lily pad on any day by $8=2^3$ to find the total size of $8$ lily pads.



                                              As this source informs, the Giant Water Lily may grow as large as $8$ to $9$ feet ($2.4-2.7$m) in diameter.






                                              share|cite|improve this answer
























                                                3












                                                3








                                                3







                                                I think that, intuitively, the answer should be less than $30/4$ since it is increasing at an exponential rate.




                                                Lily pad doubles in size every day, so it is increasing as a geometric progression.
                                                $$begin{array}{c|c|c|c|c|c|c|c|c}
                                                text{n-th day}&1&2&3&4&cdots&26&27&28&29&30\
                                                hline
                                                text{size of $1$ lily pad}&1&2&2^2&2^3&cdots&2^{25}&2^{26}&2^{27}&2^{28}&color{red}{2^{29}} end{array}$$



                                                If you start with $8$ lily pads, each doubling on its own, then:
                                                $$begin{array}{c|c|c|c|c|c|c|c|c}
                                                text{n-th day}&1&2&3&4&cdots&26&27&28&29&30\
                                                hline
                                                text{size of $8$ lily pads}&2^3&2^4&2^5&2^6&cdots&2^{28}&color{red}{2^{29}}&2^{30}&2^{31}&2^{32} end{array}$$

                                                Because when each of $8$ lily pads keeps doubling per day, the $8$ lily pads increase $8$ times faster in size altogether than that of one lily pad. So, you must multiply the size of one lily pad on any day by $8=2^3$ to find the total size of $8$ lily pads.



                                                As this source informs, the Giant Water Lily may grow as large as $8$ to $9$ feet ($2.4-2.7$m) in diameter.






                                                share|cite|improve this answer













                                                I think that, intuitively, the answer should be less than $30/4$ since it is increasing at an exponential rate.




                                                Lily pad doubles in size every day, so it is increasing as a geometric progression.
                                                $$begin{array}{c|c|c|c|c|c|c|c|c}
                                                text{n-th day}&1&2&3&4&cdots&26&27&28&29&30\
                                                hline
                                                text{size of $1$ lily pad}&1&2&2^2&2^3&cdots&2^{25}&2^{26}&2^{27}&2^{28}&color{red}{2^{29}} end{array}$$



                                                If you start with $8$ lily pads, each doubling on its own, then:
                                                $$begin{array}{c|c|c|c|c|c|c|c|c}
                                                text{n-th day}&1&2&3&4&cdots&26&27&28&29&30\
                                                hline
                                                text{size of $8$ lily pads}&2^3&2^4&2^5&2^6&cdots&2^{28}&color{red}{2^{29}}&2^{30}&2^{31}&2^{32} end{array}$$

                                                Because when each of $8$ lily pads keeps doubling per day, the $8$ lily pads increase $8$ times faster in size altogether than that of one lily pad. So, you must multiply the size of one lily pad on any day by $8=2^3$ to find the total size of $8$ lily pads.



                                                As this source informs, the Giant Water Lily may grow as large as $8$ to $9$ feet ($2.4-2.7$m) in diameter.







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                                                answered 19 hours ago









                                                farruhotafarruhota

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