Geometric ratio problem
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
add a comment |
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
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
46 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
41 mins ago
add a comment |
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
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
asked 51 mins ago
josephjoseph
4459
4459
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
46 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
41 mins ago
add a comment |
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
46 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
41 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
46 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
46 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
41 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
41 mins ago
add a comment |
4 Answers
4
active
oldest
votes
$27$ 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.
add a comment |
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.
add a comment |
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.
add a comment |
Starting with 8 pcs. is like 3 days have gone by. So 27 days remain.
add a comment |
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4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
$27$ 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.
add a comment |
$27$ 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.
add a comment |
$27$ 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.
$27$ 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.
answered 42 mins ago
ArthurArthur
111k7105186
111k7105186
add a comment |
add a comment |
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.
add a comment |
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.
add a comment |
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.
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.
answered 32 mins ago
Mostafa AyazMostafa Ayaz
14.2k3937
14.2k3937
add a comment |
add a comment |
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.
add a comment |
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.
add a comment |
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.
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.
answered 45 mins ago
Eevee TrainerEevee Trainer
5,0971734
5,0971734
add a comment |
add a comment |
Starting with 8 pcs. is like 3 days have gone by. So 27 days remain.
add a comment |
Starting with 8 pcs. is like 3 days have gone by. So 27 days remain.
add a comment |
Starting with 8 pcs. is like 3 days have gone by. So 27 days remain.
Starting with 8 pcs. is like 3 days have gone by. So 27 days remain.
answered 24 mins ago
zolizoli
16.5k41743
16.5k41743
add a comment |
add a comment |
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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
46 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
41 mins ago