Home / Mathematics /2022 Grade 12 Mathematics 7. Write neatly and legibly THE SIERPINSKI TRIANGLE RATIONALE The Mathematics department at Serena College decided to renovate its offices, and they selected a special design for the floor tiling in each office. The design is shown in the image below. The design is a fascinating pattern in mathematics, and has a special name: the Sierpinski Triangle. It is a fractal described in 1915 by Waclaw Sierpinski. It is a self-similar structure that occurs at different levels of the magnifications. The Sierpinski triangle is a fractal with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles, as seen above. It is a fascinating geometric pattern. NOTE: A fractal is a geometric construction that is self-similar at different scales. It will look the same, no matter at what size it is viewed at. To construct the Sierpinski triangle 1. Stage 0: Start with an equilateral triangle, with each side 12 cm long. (An equilateral triangle is a triangle with three equal sides and three equal angles of 60° each.). Stage 0 2 Stage 1: Copyright reserved Page 2 of 8 Please turn over Grade 12 Mathematics 2022 Locate the midpoints of each of the sides of the triangle. 3 Stage 2: Connect each of the midpoints with a line segment. This will subdivide the triangle into four smaller congruent equilateral triangles. Then remove the central one, 4. Stage 3: Repeat the three steps above for each of the remaining smaller triangles. You now have three equilateral triangles within the original equilateral triangle. Stage 1 Stage 2 Stage 3 This process can be continued indefinitely or until you choose to stop Stage 4 Repeat Use the drawings of the different stages of the Sierpinski triangle, and the information given, to perform the following activities: ACTIVITY 1 (5) 1.1 Use your Gr 11 Trig knowledge to determine the area of an equilateral triangle. Let the sides be 12 cm each. Leave your answer in simplified surd form. (3) THE ACCURACY OF THIS CALCULATION IS IMPORTANT FOR THE REST OF THE INVESTIGATION. Copyright reserved Page 3 of 8 Please turn over 1.2 Let the answer of Question 1.1 be the area of a Stage 1 Sierpinski triangle Into how many congruent triangles does the original triangle get divided for the second stage of the Sierpinski triangle fractal? (1) 1.3 What fraction of the original triangle is shaded in the second stage? (1) ACTIVITY 2 [5] 2.1 Use the answers of the previous activity to help you determine the area of the shaded (grey) par of a Stage 2 Sierpinski triangle. Write your answer in simplified surd form. (2) 2.2 Into how many shaded, congruent triangles does the original triangle get divided for the third stage of the Sierpinski triangle fractal? (1) 2.3 How many of the small congruent triangles will fit into the original triangle? (1) 2.4 What fraction of the original triangle is shaded in the third stage? (1) ACTIVITY 3 [6] Copyright reserved Page 4 of 8 Please turn over 2022 Grade 12 Mathematics 3.1 Use the answers of the previous questions to help you to determine the area of the shaded part of a Stage 3 Sierpinski triangle. Write your answer in simplified surd form. (2) 3.2 Into how many shaded, congruent triangles will the original triangle get divided for the fourth stage of the Sierpinski triangle fractal? (2) 3.3 How many of the small congruent triangles will fit into the original triangle? (1) 3.4 What fraction of the original triangle is shaded in the fourth stage? (1) ACTIVITY 4 [14] 4.1 Use the answers of the previous questions to help you to determine the area of the shaded part of a Stage 4 Sierpinski triangle. Write your answer in simplified surd form. (2) 4.2 Fill in the table below: 0 1 2 3 4 Stages Number of shaded triangles (1) (a) What pattern do you see in the numbers for the number of shaded triangles? (1) (b) Make a conjecture of the number of shaded, congruent triangles for the fifth stage of the Sierpinski triangle fractal. (1) (c) Can you build a formula for the number of shaded triangles at the n-th stage? (2) 4.3 Write the areas (correct to 1 decimal place) of each stage of the Sierpinski triangle, calculated in the previous activities as a series. Write the series to 4 terms. area of Stage 1 + area of Stage 2 + area of Stage 3 + area of Stage Copyright reserved Page 5 of 8 Please turn over

Question 2022 Grade 12 Mathematics 7. Write neatly and legibly THE SIERPINSKI TRIANGLE RATIONALE The Mathematics department at Serena College decided to renovate its offices, and they selected a special design for the floor tiling in each office. The design is shown in the image below. The design is a fascinating pattern in mathematics, and has a special name: the Sierpinski Triangle. It is a fractal described in 1915 by Waclaw Sierpinski. It is a self-similar structure that occurs at different levels of the magnifications. The Sierpinski triangle is a fractal with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles, as seen above. It is a fascinating geometric pattern. NOTE: A fractal is a geometric construction that is self-similar at different scales. It will look the same, no matter at what size it is viewed at. To construct the Sierpinski triangle 1. Stage 0: Start with an equilateral triangle, with each side 12 cm long. (An equilateral triangle is a triangle with three equal sides and three equal angles of 60° each.). Stage 0 2 Stage 1: Copyright reserved Page 2 of 8 Please turn over Grade 12 Mathematics 2022 Locate the midpoints of each of the sides of the triangle. 3 Stage 2: Connect each of the midpoints with a line segment. This will subdivide the triangle into four smaller congruent equilateral triangles. Then remove the central one, 4. Stage 3: Repeat the three steps above for each of the remaining smaller triangles. You now have three equilateral triangles within the original equilateral triangle. Stage 1 Stage 2 Stage 3 This process can be continued indefinitely or until you choose to stop Stage 4 Repeat Use the drawings of the different stages of the Sierpinski triangle, and the information given, to perform the following activities: ACTIVITY 1 (5) 1.1 Use your Gr 11 Trig knowledge to determine the area of an equilateral triangle. Let the sides be 12 cm each. Leave your answer in simplified surd form. (3) THE ACCURACY OF THIS CALCULATION IS IMPORTANT FOR THE REST OF THE INVESTIGATION. Copyright reserved Page 3 of 8 Please turn over 1.2 Let the answer of Question 1.1 be the area of a Stage 1 Sierpinski triangle Into how many congruent triangles does the original triangle get divided for the second stage of the Sierpinski triangle fractal? (1) 1.3 What fraction of the original triangle is shaded in the second stage? (1) ACTIVITY 2 [5] 2.1 Use the answers of the previous activity to help you determine the area of the shaded (grey) par of a Stage 2 Sierpinski triangle. Write your answer in simplified surd form. (2) 2.2 Into how many shaded, congruent triangles does the original triangle get divided for the third stage of the Sierpinski triangle fractal? (1) 2.3 How many of the small congruent triangles will fit into the original triangle? (1) 2.4 What fraction of the original triangle is shaded in the third stage? (1) ACTIVITY 3 [6] Copyright reserved Page 4 of 8 Please turn over 2022 Grade 12 Mathematics 3.1 Use the answers of the previous questions to help you to determine the area of the shaded part of a Stage 3 Sierpinski triangle. Write your answer in simplified surd form. (2) 3.2 Into how many shaded, congruent triangles will the original triangle get divided for the fourth stage of the Sierpinski triangle fractal? (2) 3.3 How many of the small congruent triangles will fit into the original triangle? (1) 3.4 What fraction of the original triangle is shaded in the fourth stage? (1) ACTIVITY 4 [14] 4.1 Use the answers of the previous questions to help you to determine the area of the shaded part of a Stage 4 Sierpinski triangle. Write your answer in simplified surd form. (2) 4.2 Fill in the table below: 0 1 2 3 4 Stages Number of shaded triangles (1) (a) What pattern do you see in the numbers for the number of shaded triangles? (1) (b) Make a conjecture of the number of shaded, congruent triangles for the fifth stage of the Sierpinski triangle fractal. (1) (c) Can you build a formula for the number of shaded triangles at the n-th stage? (2) 4.3 Write the areas (correct to 1 decimal place) of each stage of the Sierpinski triangle, calculated in the previous activities as a series. Write the series to 4 terms. area of Stage 1 + area of Stage 2 + area of Stage 3 + area of Stage Copyright reserved Page 5 of 8 Please turn over

T5KMTQ The Asker · Other Mathematics

Transcribed Image Text: 2022 Grade 12 Mathematics 7. Write neatly and legibly THE SIERPINSKI TRIANGLE RATIONALE The Mathematics department at Serena College decided to renovate its offices, and they selected a special design for the floor tiling in each office. The design is shown in the image below. The design is a fascinating pattern in mathematics, and has a special name: the Sierpinski Triangle. It is a fractal described in 1915 by Waclaw Sierpinski. It is a self-similar structure that occurs at different levels of the magnifications. The Sierpinski triangle is a fractal with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles, as seen above. It is a fascinating geometric pattern. NOTE: A fractal is a geometric construction that is self-similar at different scales. It will look the same, no matter at what size it is viewed at. To construct the Sierpinski triangle 1. Stage 0: Start with an equilateral triangle, with each side 12 cm long. (An equilateral triangle is a triangle with three equal sides and three equal angles of 60° each.). Stage 0 2 Stage 1: Copyright reserved Page 2 of 8 Please turn over Grade 12 Mathematics 2022 Locate the midpoints of each of the sides of the triangle. 3 Stage 2: Connect each of the midpoints with a line segment. This will subdivide the triangle into four smaller congruent equilateral triangles. Then remove the central one, 4. Stage 3: Repeat the three steps above for each of the remaining smaller triangles. You now have three equilateral triangles within the original equilateral triangle. Stage 1 Stage 2 Stage 3 This process can be continued indefinitely or until you choose to stop Stage 4 Repeat Use the drawings of the different stages of the Sierpinski triangle, and the information given, to perform the following activities: ACTIVITY 1 (5) 1.1 Use your Gr 11 Trig knowledge to determine the area of an equilateral triangle. Let the sides be 12 cm each. Leave your answer in simplified surd form. (3) THE ACCURACY OF THIS CALCULATION IS IMPORTANT FOR THE REST OF THE INVESTIGATION. Copyright reserved Page 3 of 8 Please turn over 1.2 Let the answer of Question 1.1 be the area of a Stage 1 Sierpinski triangle Into how many congruent triangles does the original triangle get divided for the second stage of the Sierpinski triangle fractal? (1) 1.3 What fraction of the original triangle is shaded in the second stage? (1) ACTIVITY 2 [5] 2.1 Use the answers of the previous activity to help you determine the area of the shaded (grey) par of a Stage 2 Sierpinski triangle. Write your answer in simplified surd form. (2) 2.2 Into how many shaded, congruent triangles does the original triangle get divided for the third stage of the Sierpinski triangle fractal? (1) 2.3 How many of the small congruent triangles will fit into the original triangle? (1) 2.4 What fraction of the original triangle is shaded in the third stage? (1) ACTIVITY 3 [6] Copyright reserved Page 4 of 8 Please turn over 2022 Grade 12 Mathematics 3.1 Use the answers of the previous questions to help you to determine the area of the shaded part of a Stage 3 Sierpinski triangle. Write your answer in simplified surd form. (2) 3.2 Into how many shaded, congruent triangles will the original triangle get divided for the fourth stage of the Sierpinski triangle fractal? (2) 3.3 How many of the small congruent triangles will fit into the original triangle? (1) 3.4 What fraction of the original triangle is shaded in the fourth stage? (1) ACTIVITY 4 [14] 4.1 Use the answers of the previous questions to help you to determine the area of the shaded part of a Stage 4 Sierpinski triangle. Write your answer in simplified surd form. (2) 4.2 Fill in the table below: 0 1 2 3 4 Stages Number of shaded triangles (1) (a) What pattern do you see in the numbers for the number of shaded triangles? (1) (b) Make a conjecture of the number of shaded, congruent triangles for the fifth stage of the Sierpinski triangle fractal. (1) (c) Can you build a formula for the number of shaded triangles at the n-th stage? (2) 4.3 Write the areas (correct to 1 decimal place) of each stage of the Sierpinski triangle, calculated in the previous activities as a series. Write the series to 4 terms. area of Stage 1 + area of Stage 2 + area of Stage 3 + area of Stage Copyright reserved Page 5 of 8 Please turn over
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Transcribed Image Text: 2022 Grade 12 Mathematics 7. Write neatly and legibly THE SIERPINSKI TRIANGLE RATIONALE The Mathematics department at Serena College decided to renovate its offices, and they selected a special design for the floor tiling in each office. The design is shown in the image below. The design is a fascinating pattern in mathematics, and has a special name: the Sierpinski Triangle. It is a fractal described in 1915 by Waclaw Sierpinski. It is a self-similar structure that occurs at different levels of the magnifications. The Sierpinski triangle is a fractal with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles, as seen above. It is a fascinating geometric pattern. NOTE: A fractal is a geometric construction that is self-similar at different scales. It will look the same, no matter at what size it is viewed at. To construct the Sierpinski triangle 1. Stage 0: Start with an equilateral triangle, with each side 12 cm long. (An equilateral triangle is a triangle with three equal sides and three equal angles of 60° each.). Stage 0 2 Stage 1: Copyright reserved Page 2 of 8 Please turn over Grade 12 Mathematics 2022 Locate the midpoints of each of the sides of the triangle. 3 Stage 2: Connect each of the midpoints with a line segment. This will subdivide the triangle into four smaller congruent equilateral triangles. Then remove the central one, 4. Stage 3: Repeat the three steps above for each of the remaining smaller triangles. You now have three equilateral triangles within the original equilateral triangle. Stage 1 Stage 2 Stage 3 This process can be continued indefinitely or until you choose to stop Stage 4 Repeat Use the drawings of the different stages of the Sierpinski triangle, and the information given, to perform the following activities: ACTIVITY 1 (5) 1.1 Use your Gr 11 Trig knowledge to determine the area of an equilateral triangle. Let the sides be 12 cm each. Leave your answer in simplified surd form. (3) THE ACCURACY OF THIS CALCULATION IS IMPORTANT FOR THE REST OF THE INVESTIGATION. Copyright reserved Page 3 of 8 Please turn over 1.2 Let the answer of Question 1.1 be the area of a Stage 1 Sierpinski triangle Into how many congruent triangles does the original triangle get divided for the second stage of the Sierpinski triangle fractal? (1) 1.3 What fraction of the original triangle is shaded in the second stage? (1) ACTIVITY 2 [5] 2.1 Use the answers of the previous activity to help you determine the area of the shaded (grey) par of a Stage 2 Sierpinski triangle. Write your answer in simplified surd form. (2) 2.2 Into how many shaded, congruent triangles does the original triangle get divided for the third stage of the Sierpinski triangle fractal? (1) 2.3 How many of the small congruent triangles will fit into the original triangle? (1) 2.4 What fraction of the original triangle is shaded in the third stage? (1) ACTIVITY 3 [6] Copyright reserved Page 4 of 8 Please turn over 2022 Grade 12 Mathematics 3.1 Use the answers of the previous questions to help you to determine the area of the shaded part of a Stage 3 Sierpinski triangle. Write your answer in simplified surd form. (2) 3.2 Into how many shaded, congruent triangles will the original triangle get divided for the fourth stage of the Sierpinski triangle fractal? (2) 3.3 How many of the small congruent triangles will fit into the original triangle? (1) 3.4 What fraction of the original triangle is shaded in the fourth stage? (1) ACTIVITY 4 [14] 4.1 Use the answers of the previous questions to help you to determine the area of the shaded part of a Stage 4 Sierpinski triangle. Write your answer in simplified surd form. (2) 4.2 Fill in the table below: 0 1 2 3 4 Stages Number of shaded triangles (1) (a) What pattern do you see in the numbers for the number of shaded triangles? (1) (b) Make a conjecture of the number of shaded, congruent triangles for the fifth stage of the Sierpinski triangle fractal. (1) (c) Can you build a formula for the number of shaded triangles at the n-th stage? (2) 4.3 Write the areas (correct to 1 decimal place) of each stage of the Sierpinski triangle, calculated in the previous activities as a series. Write the series to 4 terms. area of Stage 1 + area of Stage 2 + area of Stage 3 + area of Stage Copyright reserved Page 5 of 8 Please turn over
Community Answer
JR3BSB

Part 1.1)Part 1.2)In the second stage the triangle from stage 1 is divided into 4 congruent trianglesPart 1.3)In the second stage, 3//4 parts of the original triangle are covered.Part 2.1)Since the triangle is equilateral, then its interior angles are equal, soPart 2.2)3alpha=180^(@)rarr alpha=60^(@)In the second stage there is 1 blank triangle and 3 shaded triangles, andNow, each of the shaded triangles is divided into 4 congruent triangles,sin(60^(@))=(h)/( 12)quad rarrquad h=12 sin(60^(@))=12(sqrt3)/(2)=6sqrt3therefore in the second stage there are 3**4=12 congruent triangles, of which 3**3=9 are shadedFinally," Area "=A_(1)=(" base "**" height ")/(2)=(12**6sqrt3)/(2)=36sqrt3Part 2.3)Part 3.3)If the 4 triangles obtained in the second stage are ... See the full answer

Related Questions

People arrive at a copy machine according to a Poisson process with rate one per minute. The number of copies to be made by each person is uniformly distributed between 1 and 10. Each copy requires 3 seconds. Find the average waiting time in queue when: People with no more than 2 copies to make are given non-preemptive priority over other people. Let $class \; 1$ be people with no more than 2 copies and $class \; 2$ others. I'm having trouble calculating $E[X_1^2]$ and $E[X_2^2]$ . I'm pretty sure $ \displaystyle E[X_1] = 3 \frac{2+1}{2} = 4.5 $ and $ \displaystyle E[X_2] = 3 \frac{10+3}{2} = 19.5 $ . After this we have a waiting time formula which we plug these values into. I tried looking at $\displaystyle E[X^2] = \frac{b^2 + ab + a^2}{3}$ so get $E[X_1^2] = 7$ and $E[X_2^2] = 133$ but these don't give a reasonable answer.
Problems 3.2(e,f), 3.4(e,f), 3.5(b,c), 3.6(c,d), 3.10(a,b), 3.11, 3.12(b), 3.15(a,c) 140 Chapter 3 Gate-Level Minimization Problems 141 3.4 Simplify the following Boolean functions, using K-maps: (a) F(x,y, z) (2,3.6,7) (c)' F(A,B,C,D)=Σ(3,7, ll, 13, 14, 15) (e) F (w, x, y, z) = Σ(11, 12, 13, 14, 15) (g) F(w, x, y, z)=E(0.1.4,5,10,1 1,14,15) Simplify the following Boolean functions, using four-variable K-maps: (a)" F (w, x, y, z) = Σ(1, 4, 5, 6, 12, 14, 15) (b)" F (A, B, C, D) = Σ(2, 3, 6, 7, 12, 13, 14) (c) F (w, x, y, z) = Σ(1, 3, 4, 5, 6, 7, 9, 1 1, 13, 15) (d)' F (A, B, C, D) = Σ (0, 2, 4, 5, 6, 7, 8, 10, 13, i5) Simplify the following Boolean expressions, using four-variable K-maps: (a) A'B'C'D AC'D' B'CDA'BCD BCD (by (d)' (f) (h) F (A, B, C, D) = Σ(4. 6, 7, 15) F(w.rry, z)_E(2.3. I2. 13, 14, 15) F (w, x, y, z)-S(8, 10, 12, 13.4) F(w, x, y, z)-E(2.3.6.7.8.9.12,I3) UDP 02467 3.5 FIGURE 3.40 Schematic for Circuit with UDP_02467 3.6 Although Verilog HDL uses this kind of description for UDPs only, other HDLs and computer-aided design (CAD) systems use other procedures to specify digital circuits in tabular form. The tables can be processed by CAD software to derive an efficient gate structurc of the design. None of Verilog's predefined primitives describes sequential logic. The model of a sequential UDP requires that its output be declared as a reg data type, and that a column be added to the truth table to describe the next state. So the columns are organized as inputs: state: next state. (c) A'B'CD AB'D +A'BC ABCD AB'C (d) A'B'C'D' BC'D A'C'D A'BCD +ACD' Simplify the following Boolean expressions, using four-variable K-maps: 3.7 In this section, we introduced the HDLs and presented simple examples to illustrate alternatives for modeling combinational logic. A more detailed presentation of model- ing with HDLs can be found in the next chapter. The reader familiar with combinational circuits can go directly to Section 4.12 to continue with this subject. (c) AB C B'C'DBCD ACD'+ A'B'C+ A BC D Find the minterms of the following Boolean expressions by first plotting each function in a K-map: 3.8 (a) xy yz + xy'z (b) C'D+ABC ABD' +A'B'D (d) A'B A'CD B'CD BC'D VHDL-Truth Tables 3.9 Find all the prime implicants for the following Boolean functions, and determine which are essential: (a)" F(w, x, y, z) = Σ(0, 2, 4, 5, 6, 7, 8, 10, 13, 15) (b), F (A, B, C, D) = Σ(0, 2, 3, 5, 7, 8, 10, 11, 14, 15) (c) F (A, B, C, D) = Σ(2, 3, 4, 5, 6, 7, 9, 11, 12, 13) (d) F(w,x, y, z)=E(1.3, 6, 7, 8, 9, 12, 13, 14, 15) (e) F (A, B, C, D) = Σ(0, 1, 2, 5, 7, 8, 9, 10, 13, 15) (f) F(w, x, y, z)=E(0.1, 2, 5, 7, 8, 10, 15) Simplify the following Boolean functions by first finding the essential prime implicants: (a) F (w, x, y, z) 2(0, 2, 5, 7, 8, 10, 12, 13, 14, 15) (b) F(A, B,C, D) 2(0,2,3,5,7,8,10, 11, 14, 15) (e)* F(A, B,C,D)-(1,3,4,5, 10, 11, 12, 13, 14, 15) (d) F (w, x, y, z) = Σ(0, 1, 4, 5, 6, 7, 9, 11, 14, 15) (e) F (A, B, C, D)-Σ(0.1, 3, 7, 8, 9, 10, 13, 15) (f) F (w, x, y, z) = Σ(0, 1, 2, 4, 5, 6, 7, 10, 15) VHDL does not support truth tables directly. Instead, a truth table has to be converted into a set of Boolean equations, which can be described by signal assignment statements. PROBLEMS (Answers to problems marked with appear at the end of the text.) 3.10 1 Simplify the following Boolean functions, using three-variable K-maps: (a) F(x, y, z) = Σ (0, 2, 4, 5) (c) Fr, y, )-20,1,2,3, 5) (b) F(x,y, z)-2(0, 2, 4, 5, 6) (d) , y, z)=E(1,2,3,7) 2 Simplify the following Boolean functions, using three-variable K-maps: (b)" F(x, y, z) = Σ(1, 2, 3, 6, 7) (d) F(x, y, z) = Σ( 1, 2, 3, 5, 6, 7) (f) F(x, y, z)=E(3, 4, 5, 6, 7) (a)" F(x, y, z) = Σ (0, 1, 5, 7) (c) Fx,y, z) (2,3,4,5) (e) P(x, y, z) = Σ(0,2,4,6) Simplify the following Boolean expressions, using three-variable K-maps: (a)'. F (x.yz) = xy + x'y'z, + χ,yz (c)" F (x, y, z) = x,y + yz' + y,z' 3.11 Using K-maps for Fand F,convert the following Boolean function from a sum-of-prod- 3.3 ucts form to a simplified product-of-sums form. (d) F(x, y, z) = x,yz + xy,z' + xy'z F(w ryz)0,,2, 5,8, 10, 13)
Please help solving problem 1,2 and 3. Thank you very much. For questions 1 - 3, consider the following test results obtained for a surface water sample. lon Concentration (ppm) Ca2+ 36.0 Mg2+ 10.8 Na 12.4 K 5.9 нсоз" 82.1 SO42 32.0 CI 16.9 NO3 7.4 1. Check the accuracy of the analytical measurements. Do you suspect that a constituent of significance has been neglected? If so, is it a cation or an anion? 2. Determine the mole fraction of calcium, magnesium, and sulfate for the water sample given above. 3. Determine this water's hardness and alkalinity in mg/L as CaCO3.
Die-rolling experiment: 1) Acquire any "fair-enough" die. 2) Roll it at least 100 times (each person for themselves). 3) Note what the outcome is overy time. 4) For each group of 5 (go consecutively, i.e. group first five rolls, then the next five rolls, etc.), take the average of the 5 rolls. For example, if the first five rolls are \( 3,6,1 \), 5 , and 3 , Die-rolling experiment: 1) Acquire any "fair-enough" die. 2) Roll it at least 100 times (each person for themselves). 3) Note what the outcome is overy time. 4) For each group of 5 (go consecutively, i.e. group first five rolls, then the next five rolls, etc.), take the average of the 5 rolls. For example, if the first five rolls are \( 3,6,1 \), 5 , and 3 , the average will be \( 18 / 5=3.6 \). 5) You will end up with at least 20 averages. Plot these averages on a \( 2 \mathrm{D} \) column graph. Alternatively, you can do a histogram of your results, which will be a fancier way of doing a 2D column graph. With a histogram though, you can control the bin size and instead of each individual average (e.g. 3.2, 3.4. 3.6, etc.). you can group them together with a bin size of 1 . Bonus credits will be assigned to those showing their results with a histogram of bin size 1 in addition to their regular \( 2 \mathrm{D} \) column graphs. 6) All the probabilities of \( 1,2,3,4,5 \) and 6 are \( 1 / 6 \) with a fair die. Therefore, the probabilities are said to be "uniformly distributed". Then, why is your \( 2 D \) column graph / histogram not looking uniform at all? Speculate / comment on the rationale for this.
2-4 Estimate the TDS for one of the water samples in Problem \( 2-1 \) or \( 2-2 \) (to be selected by instructor) using Eq. (2-11) and by summing the mass of the individual ionic species. How do the computed values compare?The following test results were obtained for three treated wastewater effluent samples. Check the accuracy of the analytical 2-4 Estimate the TDS for one of the water samples in Problem \( 2-1 \) or \( 2-2 \) (to be selected by instructor) using Eq. (2-11) and by summing the mass of the individual ionic species. How do the computed values compare?The following test results were obtained for three treated wastewater effluent samples. Check the accuracy of the analytical measurements for one of the waters (to be selected by instructor). Do you suspect that a constituent of significance has been neglected? If so, is it a cation or anion? Determine the mole fraction of \( \mathrm{Ca}^{2+}, \mathrm{Mg}^{2+} \), and \( \mathrm{SO}_{4}^{2-} \) for one of the following treated wastewater effluents (to be selected by instructor).
Which of the following statements is correct regarding Kepler's three laws of planetary motion? Angular momentum of any planet about an axis through the Sun and perpendicular to the orbital plane is not invariant during its motion. Planets move in elliptical orbits with the sun not being at its center. Orbital period of a planet is directly proportional to Which of the following statements is correct regarding Kepler's three laws of planetary motion? Angular momentum of any planet about an axis through the Sun and perpendicular to the orbital plane is not invariant during its motion. Planets move in elliptical orbits with the sun not being at its center. Orbital period of a planet is directly proportional to its distance from the sun. Kepler's laws are valid only for planets in the solar system.
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