Assignment 07

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course MTH 151

8:50pm, 2/18/14

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

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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?

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Your solution:

The first triangle would have 3 points. The second triangle would have 6 points, because the lengths of the segments AB and AC are doubled, and by connecting the endpoints to form a new triangle, we get 6 points all together.

confidence rating #$&*: 3

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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:

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Self-critique (if necessary):

I could see how this could become somewhat confusing. I myself had to draw it out, and really be careful with my points and counting them to get an accurate number.

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Self-critique Rating: 3

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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?

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Your solution:

There are 10 points total in the figure. By extending the two sides, 2 points are added to each side, making 4 points. These 4 points added to the 6 original points make 10.

confidence rating #$&*: 3

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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:

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Self-critique (if necessary):

OK

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Self-critique Rating: 3

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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?

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Your solution:

When extending each side one step further, each side will have 5 points, making the total triangle have 15 points total.

confidence rating #$&*: 3

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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

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Self-critique (if necessary):

OK

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Self-critique Rating: 3

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

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Your solution:

When extending the process one more step, there would be 18 points, because each side would have 6 points.

confidence rating #$&*: 3

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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.

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Self-critique (if necessary):

I didn’t add 4 spaced points to the new side, I just added one more point to each side.

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Self-critique Rating: 3

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

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Your solution:

The sequence seems to follow the sequence + 3, + 4, +5, +6 because 3 + 3 = 6, 6 + 4 = 10, 10 + 5 = 15, 15 + 6 = 21. According to this pattern, the next number should be 28, because 21 + 7 = 28.

confidence rating #$&*: 3

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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.

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Self-critique (if necessary):

OK

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Self-critique Rating: 3

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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?

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Your solution:

Adding another point to the sides of the triangle would make each side of the triangle have 7 points, and the number 7 comes next in our sequence to be added.

confidence rating #$&*: 3

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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.

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Self-critique (if necessary):

OK

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Self-critique Rating: 3

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

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Your solution:

Adding an endpoint to each side of the triangle increases the number of endpoints by 1 each time.

confidence rating #$&*: 3

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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.

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Self-critique (if necessary):

OK

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Self-critique Rating: 3

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.

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

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Your solution:

There should be 72 BB’s total in the above picture, because 24 * 3 = 72. Removing the top half should give us 12 BB’s on each side, so 12 * 3 = 36, with 36 BB’s total if the top half was removed.

confidence rating #$&*:

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

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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?

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Your solution: 776 / 2 = 388. 388 / 3 (sides of triangle) = 129. 3.

@&

The BB's are not distributed just over the sides of the triangles.

If adding a row adds 51 BB's, then the since the first row contains 1 BB and each additional row contains 1 more than the preceding, the number of rows must be 51, give or take 1. What would be the exact number, and why?

*@

confidence rating #$&*:

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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?

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Your solution: T_14 = ½ * 14 * (14+1) = 105; T_15 = ½ * 15 * (15+1) = 120

confidence rating #$&*:

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Self-critique Rating:

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