help please w/ word problem: "Jane started to knit a blanket...."

Jane started to knit a blanket. This blanket consists of knitted flowers. The diameter of one flower is 10 cm. The size of the blanket is 250cmX260cm. On the first day she knitted two flowers. How long does it take to knit the blanket (before combining the flowers) if she would always knit on the next day

a) two more flowers
b) two times more than in the previous day

jms said:

Jane started to knit a blanket. This blanket consists of knitted flowers. The diameter of one flower is 10 cm. The size of the blanket is 250cmX260cm. On the first day she knitted two flowers. How long does it take to knit the blanket (before combining the flowers) if she would always knit on the next day

a) two more flowers
b) two times more than in the previous day

Click to expand...

How much time does it take to fill in the space between circles ("flowers")? And how are the circles arranged?

When you reply, please include a clear listing of your steps and thoughts so far. Thank you!

So do I have to know how much space there is between the flowers? I would start to do this by calculating the area of one flower A=π.r^2= π*5^2

And then the area of the whole blanket A=250*260cm and to get the ratio of how many flowers the blanket have. But I'm not getting the number of how many flowers the blanket can have?

I'm just really lost with this

Hello, jms!

Jane started to knit a blanket of knitted flowers.
The diameter of one flower is 10 cm.
The size of the blanket is 250cm x 260cm.
On the first day she knitted two flowers.
How long does it take to knit the blanket (before combining the flowers)
if she would always knit on the next day:

a) two more flowers than in the previous day.
b) two times more than in the previous day.

Click to expand...

stapel has a legitimate question.

If they are arranged like this: .\(\displaystyle \begin{array}{c} \bigcirc \bigcirc \bigcirc \\ \bigcirc \bigcirc \bigcirc \\ \bigcirc\bigcirc\bigcirc \end{array}\)

. . there will be: .\(\displaystyle 25 \times 26 \,=\,650\) flowers.


If they are arranged like this: .\(\displaystyle \begin{array}{c}\bigcirc\bigcirc\bigcirc\bigcirc \\ \bigcirc\bigcirc\bigcirc \\ \bigcirc \bigcirc\bigcirc\bigcirc \end{array}\)

. . there will be more flowers.

hmm it really doesn't say anything else, so im assuming there are no space between the flowers, rather they make the blanket.

jms said:

hmm it really doesn't say anything else, so im assuming there are no space between the flowers, rather they make the blanket.

Click to expand...

If the flowers were, say, squares, then you'd have no space between them, because you can easily line them up to join along all their edges. But you've been given that the flowers are circles. There must be space between them, since they can't match up all along their (circular) sides.

You may need to ask your instructor for clarification.

So the instructor said the blanket will have as many flowers as it can have and told me not making that bit too difficult as the main point is creating an equation, and this should be very simples, not for me though.

jms said:

Jane started to knit a blanket. This blanket consists of knitted flowers. The diameter of one flower is 10 cm. The size of the blanket is 250cmX260cm. On the first day she knitted two flowers. How long does it take to knit the blanket (before combining the flowers) if she would always knit on the next day

a) two more flowers

Click to expand...

This, also, is ambiguous. It could mean that she knitted two flowers a day- so she knitted "two more flowers" every day- or it could mean that she knitted "two more flowers" than she had the previous day.
The blancket is 250 by 250 cm so 62500 square cm. The flowers are 10 cm in diameter, so 5 cm in radius so \(\displaystyle \pi(5)^2= 25\pi= 78.5 square centimeters. But, as others have pointed out, circles won't fill a square. There are two ways to handle that- (1) ignore the area about them- there are 62500/78.5= 795.8 or 796 roses or (2) assume the roses are actually knitted on 10 by 10 (100 square cm) squares so there are 62500/100= 625 squares. It is probably the latter that is intended.

If she knitted two roses a day, it would take 625/2= 312.5 days.

If she knitted two more roses than the previous day the number or roses knitted is an "arithmetic series": 2+ 4+ 6+ 8+ ...= 2(1+ 2+ 3+ 4+...)
Notice 2 times the sum is the same as adding the sum to itself and that if we add (1+ 2+ 3+ 4+ ...+ (n-1)+ n) to the reversed sum (n+ (n-1)+ ... + 4+ 3+ 2+ 1) "term by term", each term adds to n+2: (n)+ (1), (n-1)+ 2, (n-2)+ 3, etc. are all equal to n+1. Since there are n terms, that is n(n+1) total roses knitted in n days.
So either n(n+1)= 625. That is the same as the quadratic equation \(\displaystyle n^2+ n- 625= 0\). That has two solutions but only one, 24.5 days, is positive.

b) two times more than in the previous day

Click to expand...

This is a "geometric series": \(\displaystyle 2+ 4+ 6+ 8+ ...+ 2^n\) If we write \(\displaystyle S= 2+ 4+ 6+ 8+ ...+ 2^n\) then we can see that \(\displaystyle S- 2= 4+ 6+ 8+ ...+ 2^n= 2(2+ 3+ 4+ ...+ 2^{n-1})\). If we add \(\displaystyle 2^{n+1}= 2(2^n)\) to both sides, it becomes \(\displaystyle S- 2+ 2^{n+1}= 2(2+ 3+ 4+ ...+ 2^n)\) or \(\displaystyle S- 2+ 2^{n+1}= 2S. Subtracting S from both sides, \(\displaystyle S= 2^{n+1}- 2\).

That is, we must have \(\displaystyle 2^{n+1}- 2= 625\) or \(\displaystyle 2^{n+1}= 627\). Of course, \(\displaystyle 2^9= 512 and \(\displaystyle 2^10= 1024\) so this should require 10 days (9 days and a little work on the 10th day.)\)\)\)