Problem Statement:
1. A gardener is trying to figure out
how to water two flowers equally. The equipment that she has is one sprinkler
that sprays water in a circle. However, the closer the flower is to the center
of the sprinkler, the more water the flower receives. The problem is asking us
to figure out where to place the sprinkler so that the two, previously planted
flowers, will receive the same amount of water.
2. Now the gardener is trying to place
the sprinkler in a spot where three flowers are able to be watered equally
a. Is it always possible for three
flowers to be watered by one circular sprinkler?
b. Find the places where it is possible
for three flowers to be watered by one sprinkler.
3. Is this still possible with four,
five, or six flowers? How can we generalize the statement?
Process:
1. When I was first setting up this
problem, I created a situation that could warm my brain up to what exactly was
being asked of me. This consisted of strategically placing two flowers an even
amount of units apart. In this case, I chose to place Flower A 12 units away
from Flower B. I decided to use an even amount of units so that I would be able
to visually set what directly half of those units would look like.
2. Once I laid those two points, I
decided to count each unit until I reached six. This is where I placed my
sprinkler. I place my sprinkler here because I knew that 6 units is directly
half of my even amount of units, 12. Therefore, my circular sprinkler would be
able to reach Flower A and Flower B with the exact same amount of water.
3. After I did this warm up problem and
was visually able to see my results, I decided to try a problem that would not
be as easily solved. What would the problem look like if the two flowers were
not placed and even amount of units apart?
4. I then randomly drew two dots on my
paper to represent the two planted flowers. I knew that I would not be able to
come up with an exact midsection of these two points by just eyeballing it, so
I had to think of another way. I decided to use my compass, through
constructions, in order to figure out where the half-way point is of these two
flowers
5. My first step was to connect the two
flowers with a single blue line. This line represents the distance between
Flower A and Flower B. I have to find the midpoint of this line in order to
figure out where to place my sprinkler.
6. In order to do this, I took my
compass and opened it up so that it was wider than what I thought would be half-way.
I placed the tip of my compass on Flower A and then drew a semi-circle between
the two flowers. Without changing the width of my compass, I repeated the same
step starting on Flower B.
7. This created two points where the two
semi-circles overlapped. I colored these two points in with a red dot and
connected them with a red line. Where my red and blue line intersected was the
half-way point for my blue line.
8. I colored the intersection of these
lines with a black dot. I was then able to place the sprinkler there in order
to water each flower with the same amount of liquid. I does not matter how
large the radius of the sprinkler is, as long as it is able to reach both
flowers, the amount of water will be the same at this point.
9. Next, I repeated the same process
with three flowers. I started off by again placing three random black dots on
the grid paper to represent each flower. However, instead of just finding the
midway point between two flowers, I had to find the intersection of all three
midpoints between the flowers.
10. I was able to do this by first
drawing a blue line from each flower to the next. This created a triangle. I
then used my compass to find a bisector for each of the three blue lines using
constructions. These lines were then colored red. Once I found the bisector, I
extended each red line until all three lines intersected. At this point of
intersection is where I would place my sprinkler. I colored an orange circle to
represent the circumference of the sprinkler. This is where all three flowers
would be watered equally.
11. Now that I was able to figure out a
way where three flowers were possible for one sprinkler to water each equally,
I had to next figure out if this is always the
case with three flowers.
12. After reviewing my previous example
of the three flowers, I noticed that all three points did not lie on a single
line. From here I wondered if all three flowers were collinear, would one
single curricular sprinkler water each flower equally?
13. I counted out three flowers that were
equidistance apart from one another. I quickly realized that this case would be
impossible. The sprinkler would have to be placed at Flower B in order to water
Flowers A and C equally. This would mean that Flower B would receive more water
than the other two. Therefore, three or more flowers that are collinear would
not be able to fit this problem.
14. The last section of this problem of
the week is used to generalize this situation with multiple flowers. I decided
to repeat my process from the previous two parts using four flowers. I randomly
placed four dots on the grid paper to represent the four flowers. From here, I
drew blue lines connecting the flowers and bisected each line using my compass
as before.
15. However, I was not able to come up with a
solution as to where to put my sprinkler in order to water all four flowers
equally. In order to figure out if this is the case for all flowers greater
than three, I decided to not stop there.
16. I next created a situation of where I
would be able to plant four flowers where they would all be able to obtain
water equally. In order to do this, I set up four dots representing flowers into
a square. Since I would be easily able to cut the square into equal parts, I
did not have to use my compass. From here, I was able to see how I would be
able to water four flowers equally with one sprinkler.
17. By placing the sprinkler in the
center of the square, it would be able to reach all four flowers with the same
amount of water.
18. So, what if there were more than four
flowers? Since I was able to figure out the placement for the flowers when
there were four, I decided to look deeper into five, six, and seven flowers.
When it came to the four flowers, each one laid on one circle.
19. Using this information, I was able to see that
it did not matter how many flowers were planted, but they would have to each
lie on the same circle in order to be watered equally by one circular
sprinkler.
Solution: There are different solutions to each
step of the problem, but they are all related. You are able to figure out a
spot to place a single curricular sprinkler in order to water as many flowers
as you want. However, they are a few constrictions.
1. Two flowers- endless possibilities.
As long as it is physically capable for the radius of the sprinkler to reach
each flower, you are able to place it in a spot where both flowers receive the
same amount of water.
2. Three flowers- If the three flowers
are not collinear, you will be able to find a spot for one circular sprinkler
to water each flower equally. However, if they are all in a single line, this
would be impossible.
3. Four or more- When it comes to four
or more flowers, it is still possible for each to get watered equally by one circular
sprinkler. Yet, each of the flowers would have to lie on one single circle.
Since the flowers are already planted, it is unlikely that this could be accomplished
without previous knowledge of the sprinkler.
