Geometric ratio problem












1















A lily pad sits on a pod. 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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  • 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
    36 mins ago






  • 2




    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
    30 mins ago


















1















A lily pad sits on a pod. 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







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
    36 mins ago






  • 2




    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
    30 mins ago
















1












1








1








A lily pad sits on a pod. 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







share|cite|improve this question














A lily pad sits on a pod. 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




sequences-and-series algebra-precalculus






share|cite|improve this question













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asked 40 mins ago









josephjoseph

4409




4409








  • 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
    36 mins ago






  • 2




    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
    30 mins 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
    36 mins ago






  • 2




    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
    30 mins 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
36 mins 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
36 mins ago




2




2




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
30 mins 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
30 mins ago












4 Answers
4






active

oldest

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2














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





























    2














    $27$ is the right answer. Consider at which point the original lily pad is 8 times its original size (after the 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





























      2














      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.






      share|cite|improve this answer





























        1














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






        share|cite|improve this answer





















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          4 Answers
          4






          active

          oldest

          votes








          4 Answers
          4






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          2














          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


























            2














            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
























              2












              2








              2






              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 34 mins ago









              Eevee TrainerEevee Trainer

              5,0971734




              5,0971734























                  2














                  $27$ is the right answer. Consider at which point the original lily pad is 8 times its original size (after the 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


























                    2














                    $27$ is the right answer. Consider at which point the original lily pad is 8 times its original size (after the 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
























                      2












                      2








                      2






                      $27$ is the right answer. Consider at which point the original lily pad is 8 times its original size (after the 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 the 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










                      answered 32 mins ago









                      ArthurArthur

                      111k7105186




                      111k7105186























                          2














                          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.






                          share|cite|improve this answer


























                            2














                            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.






                            share|cite|improve this answer
























                              2












                              2








                              2






                              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.






                              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.







                              share|cite|improve this answer












                              share|cite|improve this answer



                              share|cite|improve this answer










                              answered 22 mins ago









                              Mostafa AyazMostafa Ayaz

                              14.2k3937




                              14.2k3937























                                  1














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






                                  share|cite|improve this answer


























                                    1














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






                                    share|cite|improve this answer
























                                      1












                                      1








                                      1






                                      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.







                                      share|cite|improve this answer












                                      share|cite|improve this answer



                                      share|cite|improve this answer










                                      answered 13 mins ago









                                      zolizoli

                                      16.4k41743




                                      16.4k41743






























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