assignment 07 qa

#$&*

course MTH 151

If your solution to stated problem does not match the given solution, you should self-critique per instructions at

http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm

.

Your solution, attempt at solution. If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.

007. Triangular, Square, Pentagonal Numbers

*********************************************

Question: `q001. Note that there are 10 questions in this assignment.

Sketch three points A, B and C forming an equilateral triangle on a piece of paper, with point A at the lower left-hand corner, point B at the lower right-hand corner and point C at the top. Sketch the segments AB and AC.

Now double the lengths of AB and AC, and place a point at each of the endpoints of these segments. Connect these new endpoints to form a new equilateral triangle. Two sides of this triangle will have three points marked while the new side will only have its two endpoints marked. Fix that by marking that middle point, so all three sides of your new triangle are marked the same.

How many marked points were there in the original triangle, and how many are there in the new triangle?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

There are 3 points (A, B, C) in the original triangle. There are 6 points in the new triangle, the 3 from the original then the two at the end of the new segments and one in the middle of the line you drew to connet these endpoints.

confidence rating #$&*: 3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

The original triangle had the three points A, B and C. When you extended the two sides you marked the new endpoints, then you marked the point in the middle of the third side. So you've got 6 points marked.

The construction of these numbers is shown in the figure below. We begin with a single dot:

We label this point A and construct a triangle containing this point as a vertex. We place similar dots at the vertices of this triangle.

We now 'scale up' the triangle by doubling the lengths of its sides:

We divide this triangle into triangles of the original size, and place dots at each of these vertices.

The first figure has a single 'dot', the second has 3 'dots', and the third has 6 'dots'.

Note the similarity with the figures below.

The first depicts the pattern illustrated in this question.

The second illustrates the pattern extended one steps:

The third depicts the pattern as it would appear if extended 12 steps beyond the original triangle:

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):ok

------------------------------------------------

Self-critique Rating:ok

*********************************************

Question: `q002. Extend the two sides that meet at A by distances equal to the original lengths AC and AB and mark the endpoints of the newly extended segments. Each of the newly extended sides will have 4 marked points. Now connect the new endpoints to form a new right triangle. Mark points along the new side at the same intervals that occur on the other two sides. How many marked points are on your new triangle, and how many in the whole figure?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

There are 9 points on this triangle and 11 on the whole figure. when you extend the segments you add points to the end of those giving you two more points. then you connect the segments to form a new triangle, you add two points to this line so that it has the same number of points as the other lines. adding the 6 points we adlready had this gives us 10 points. however, when counting points for the new triangle only, you do not count the point in the middle of this new triangle, so you only have 9 points.

confidence rating #$&*: 3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

You added the two new endpoints when you extended the sides. You then should have marked two new points on the new third side, so that each side contains 4 points including its endpoints. Your figure will now contain 10 marked points.

The construction is shown below. First we extend the two sides by a length equal to that of the original triangle:

Next we join the 'free' endpoints of those new sides to form a triangle.

Now we place points along the new side and join them to complete the 'small' triangles within our new figure:

We have added four new dots.

The figure below depicts only the 'dots', without the triangles:

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):ok

------------------------------------------------

Self-critique Rating:ok

*********************************************

Question: `q003. Continue the process for another step-extend each side by a distance equal to the original point-to-point distance. How many points do you have in the new triangle?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

this new triangle has 12 points. you have two new points when you extend the segments. this makes each of these lines have 5 points. you add another 3 points when you connect the two segments to form a triangle so that this side has equal points to the other sides. this gives you a total of 12 points since three of these points are counted twice. There are three other points that we have added within this figure, two for the previous triangle, and 1 on the triangle before that.

confidence rating #$&*: 3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

You will add an endpoint to each newly extended side, so each of the new sides will contain 5 points. You will then have to add 3 equally spaced points to the new side, giving you a total of 13 points on the new triangle. In addition there are two marked points inside the triangle, for a total of 15 points.

Click on 'Next Picture' to see the construction. The line segments along two sides of the triangle have again been extended and points marked at the ends of these segments. The new endpoints have been connected to form the third side of a larger triangle, and equally spaced points have been constructed along that side.

`routine triangle4

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):

You say that there are 13 points but i drew this out and counted them, making sure not to count that first point twice and there are only 12 points. you can also come to this conclusion without simply counting the points on the drawn triangle. if there are 5 points on 3 sides that is 15 points. however, 3 of those points lay on the corners and would be counted by two sides. so you have to subtract the number of points that are being counted twice. 3*3=15-3=12

also you state that there are only 2 points in the figure that are not in this triangle, however this is the third time extending the triangle. for the first time we had to add one point and for the second time we had to add two points so this would be a total of 3 points that are not in the current trianlge but are in the figure.

------------------------------------------------

Self-critique Rating:3

@&

When you extend the line you get 2 new points at the ends of the extensions.

There will also be a line connecting these endpoints, with an additional 3 points between the endpoints.

Another way to look at this:

The first triangle has 3 points.

The first extension of the triangle has 3 points, giving you 6 points.

The second extension has 4 points, giving you 10 points.

The third extension, the one made here, has 5 points (the two endpoints and the 3 between the ends), for a total of 15.

*@

*********************************************

Question: `q004. Continue the process for one more step. How many points do you have in the new triangle?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

this new triangle has 15 points. each new line will have 6 points after you place the points on the end of the segments and add the 4 points needed on the line that connects the lines to form a new trianlge. three of these points are counted twice. 6*3=18-3=15

when you add the points on the lines inside this new triangle you get 5 points 1+2+3=6

15+6=21 total points in the whole figure

confidence rating #$&*: 3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

You will add an endpoint to each newly extended side, so each of the new sides will contain 6 points. You will then have to add 4 equally spaced points to the new side, giving you a total of 15 points on the new triangle. There are also 5 marked points inside the triangle for a total of 21 marked points.

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):

we have the same number of points for the new triangle and for the whole figure. however, your addition does not add up. 15+5 does not equal 21. did you mean to say that there are 6 points inside of the triangle?

------------------------------------------------

Self-critique Rating:3

*********************************************

Question: `q005. The sequence of marked points is 3, 6, 10, 15, 21. What do expect will be the next number in this sequence?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

the next number would be 28

you are adding +3,+4,+5,ect.. so the next would be +7

21+7=28

confidence rating #$&*: 3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

The differences between these numbers are 3, 4, 5, 6. The next difference, according to this pattern, should be 7, which would make the next number 28.

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):ok

------------------------------------------------

Self-critique Rating:ok

*********************************************

Question: `q006. How can you tell, in terms of the process you used to construct these triangles, that the next number should be 7 greater?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

you will add two new points to the end of the segments and then you will add 5 points onto the line that connects these lines to form the new triangle. this gives us the 7 new points.

confidence rating #$&*: 3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

When you extend the triangle again, you will add two new endpoints and each side will now have 7 points. The 7 points on the new triangle will be all of the new points.

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):ok

------------------------------------------------

Self-critique Rating:

*********************************************

Question: `q007. How do you know this sequence will continue in this manner?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

each time you are adding 1 more than in the previous extention. since counting numbers continue, you can continue adding 1 to the previous extention

confidence rating #$&*: 3

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

.............................................

Given Solution:

Each time you extend the triangle, each side increases by 1. All the new marked points are on the new side, so the total number of marked points will increase by 1 more than with the previous extension.

&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&

Self-critique (if necessary):

------------------------------------------------

Self-critique Rating:ok

The picture below depicts this sequence extended to the 24th number.

You should understand why the number of beads in this picture is 1 + 2 + 3 + 4 + 5 + ... + 21 + 22 + 23 + 24.

*********************************************

Question: `q008. How many BB's would there be if the top half of the triangle in the above picture was removed?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

if you remove the top half there would only be 12 rows. 1+2+3+4+5+6+7+8+9+10+11+12=78

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

The last two questions are a little more challenging than most q_a_ questions. See if you can get them.

*********************************************

Question: `q009. A triangle similar to the above triangle, but containing about twice as many rows, would contain 725 BB's. If another row is added there will be 776 BB's. How many rows are there in each of the triangles?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

I don't understand this problem. I've tried to work it out but i'm not getting it. is BB's the number of beads? going by everything we've leanred if you can find the number of beads by adding 1+2+3+4,... then if you continue doing this you do not end up on 725 or 776. also if you go by the difference in between the two rows, that is 51, which would suggest that 776 is the 51st row. but +1,+2,+3,...,+51=1326

if you work the formula you provide in the next question you can find that row 37 has 703 beads, 38 has 741 beads and 39 has 780 beads

t_37=1/2 * 37 * (37+1) = 18.5 * 38 = 703

t_38=1/2 * 38 * (38+1) = 19 * 39 = 741

t_39=1/2 * 39 * (39+1) = 19.5 * 40 = 780

@&

Your solution is good.

*@

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

*********************************************

Question: `q010. The formula

T_n = 1/2 * n * (n+1)

can be used to find the number of BB's in a triangle with n rows (this number is called the nth triangular number).

So for example the fourth triangular number is

T_4 = 1/2 * 4 * (4 + 1) = 10.

The sequence 1, 3, 6, 10, 15, 21, ... of triangular numbers could be written in symbols as T_1, T_2, T_3, T_4, T_5, T_6, ... .

Use the formula to find the 14th and the 15th triangular numbers, and calculate the difference between them. Why did you get the difference you did?

YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY

Your solution:

T_n=1/2*14*(14+1)

t_n=7*15

t_n = 105

t_n=1/2*15*(15+1)

t_n=7.5*16

t_n = 120

120-105=15

there is 15 difference between these rows. this is because each row you are adding 1 more than you did in the previous row. the second row you add 2 points, the third you would add 3 points, 4th is 4 points and so forth so on the 15th row you would add 15 points to your previous amount

confidence rating #$&*:

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

------------------------------------------------

Self-critique Rating:ok

&#Good work. See my notes and let me know if you have questions. &#