## Reflection

This year, math was difficult, as it is every year for me. I can say that I learned about matrices, combinations/permutations, and a bit about solving linear equations. I just couldn’t find much interest in the class, and many times when I tried to pay attention, I couldn’t absorb the information. John was probably one of the nicest math teachers I’ve had, so I do feel bad that I didn’t seem very engaged in the class. All of the POWs, but one, had a decent amount of class time to do and were somewhat understandable.

## Unit Problem

A circle with a radius of 50 has smaller circles(“trees”) on every intersection of the graph within the larger circle except for the singular midpoint. Each unit of this graph represents 10 feet. These “trees” grow about 1.5 square inches in area in a year, they’re already 2.5 inches in diameter. If one were to draw a line from the midpoint of the large circle to the point (25,0.5) of the graph, how long in years or days would it take for the “trees” to envelop, or at least touch, the drawn line?

1.5 = 3.14r^2 r = 0.7

2.5 = 2 * 3.14r r = 1.6

1.6/0.7 = 2.3 (It has already been 2.3 years)

10 * 12/0.7 = 171.4

171.4 - 2.3 = 169. 1

169.1/2 = 84.6

2/490 * 84.6 = 0.3 (2/490 was an estimation)

0.3 * 365 = 126

First, I took both the “1.5 square inches per year” and “circumference of 2.5 inches”, and brought them to be radiuses. Then I turned the radius per year into inches and took away what had already grown, using half of that with an estimation of how far the line would be from its nearest tree to figure out the fraction of a year it would take for the tree to get to that point.

It would take 126 days for the radius of a “tree” to reach the drawn line.

This problem was confusing, but the only part that I’m not entirely confident about was the 2/490 estimation.

1.5 = 3.14r^2 r = 0.7

2.5 = 2 * 3.14r r = 1.6

1.6/0.7 = 2.3 (It has already been 2.3 years)

10 * 12/0.7 = 171.4

171.4 - 2.3 = 169. 1

169.1/2 = 84.6

2/490 * 84.6 = 0.3 (2/490 was an estimation)

0.3 * 365 = 126

First, I took both the “1.5 square inches per year” and “circumference of 2.5 inches”, and brought them to be radiuses. Then I turned the radius per year into inches and took away what had already grown, using half of that with an estimation of how far the line would be from its nearest tree to figure out the fraction of a year it would take for the tree to get to that point.

It would take 126 days for the radius of a “tree” to reach the drawn line.

This problem was confusing, but the only part that I’m not entirely confident about was the 2/490 estimation.

## POW 1

1. Two delicate flowers were planted in a garden. The gardener,

Leslie, has a sprinkler that sprays water around in a circle. The closer

a flower is to the sprinkler, the more water it gets.

To be sure that her flowers each get the same amount of water,

Leslie needs to place the sprinkler where it will be the same distance

from each of the flowers.

What are her choices about where to put the sprinkler? Describe all the possibilities.

(Reminder: The flowers are already in place, and Leslie needs to adjust the position of the

sprinkler relative to the flowers.)

2. Now suppose Leslie plants three flowers and wants to know if it will still be possible to

place the sprinkler the same distance from all three.

a. Determine which arrangements of the flowers (if any) will make this possible and

which (if any) will make it impossible. (As in Question 1, Leslie will be looking for a

place to put the sprinkler after the flowers have already been planted.)

b. For those arrangements for which it will be possible, describe how Leslie can find the

correct location (or locations) for the sprinkler.

3. What about four flowers? Five flowers? Generalize as much as you can.

Your POW is to explain as fully as possible, for various cases, where Leslie can put the

sprinkler in order to give the flowers the same amount of water. Homework 2: Only Two

Flowers gets you started with the first question of the POW.

Leslie, has a sprinkler that sprays water around in a circle. The closer

a flower is to the sprinkler, the more water it gets.

To be sure that her flowers each get the same amount of water,

Leslie needs to place the sprinkler where it will be the same distance

from each of the flowers.

What are her choices about where to put the sprinkler? Describe all the possibilities.

(Reminder: The flowers are already in place, and Leslie needs to adjust the position of the

sprinkler relative to the flowers.)

2. Now suppose Leslie plants three flowers and wants to know if it will still be possible to

place the sprinkler the same distance from all three.

a. Determine which arrangements of the flowers (if any) will make this possible and

which (if any) will make it impossible. (As in Question 1, Leslie will be looking for a

place to put the sprinkler after the flowers have already been planted.)

b. For those arrangements for which it will be possible, describe how Leslie can find the

correct location (or locations) for the sprinkler.

3. What about four flowers? Five flowers? Generalize as much as you can.

Your POW is to explain as fully as possible, for various cases, where Leslie can put the

sprinkler in order to give the flowers the same amount of water. Homework 2: Only Two

Flowers gets you started with the first question of the POW.

2)

Place flowers in a triangle and sprinkler in the very middle

3)

4 flowers: Place flowers in a square

5 flowers: Place flowers in a pentagon

6 flowers: Place flowers in a hexagon

7 flowers: Place flowers in a heptagon

8 flowers: Place flowers in an octagon

9 flowers: Place flowers in a nonagon

10 flowers: Place flowers in a decagon

… and so on, flowers being clustered closer together the more there are in a roughly circular shape, the sprinkler always being in the middle with flowers as close as they can get without infringing on each other’s livelihood. Note that with this tactic, the more flowers there are, the less water they’d all get.

Also, can there be more than one sprinkler? If so, the flowers can be spaced farther and more can be put in this garden. Think, many small shapes of flowers around farther apart sprinklers. And depending on the type of sprinkler, the shapes would have to be greatly different in form and size, but I‘d rather not get into that.

In order to know exactly how far apart these flowers would need to be from each other (not to mention from the sprinkler, to avoid overwatering), information about what type of flowers they are would need to be provided.

Place flowers in a triangle and sprinkler in the very middle

3)

4 flowers: Place flowers in a square

5 flowers: Place flowers in a pentagon

6 flowers: Place flowers in a hexagon

7 flowers: Place flowers in a heptagon

8 flowers: Place flowers in an octagon

9 flowers: Place flowers in a nonagon

10 flowers: Place flowers in a decagon

… and so on, flowers being clustered closer together the more there are in a roughly circular shape, the sprinkler always being in the middle with flowers as close as they can get without infringing on each other’s livelihood. Note that with this tactic, the more flowers there are, the less water they’d all get.

Also, can there be more than one sprinkler? If so, the flowers can be spaced farther and more can be put in this garden. Think, many small shapes of flowers around farther apart sprinklers. And depending on the type of sprinkler, the shapes would have to be greatly different in form and size, but I‘d rather not get into that.

In order to know exactly how far apart these flowers would need to be from each other (not to mention from the sprinkler, to avoid overwatering), information about what type of flowers they are would need to be provided.