Project 4.1 Puzzle Design Challenge

Hello, I am designer in training learning at Arcadia High School. I enjoy nature and the city. In free time, I like to play video games and read books.

Activity 4.1a Puzzle Cube Combinations

1.         Why is it so important for a designer to think of multiple solutions to a design problem?

 Multiple solutions allows designers to test a variety of designs. More over he or she has a back up design. In addition hybrid designs from previous solutions may arise.

2.         What steps did you take to determine the exact number of possible combinations for each set of cubes?

 I start of by only moving a cube and testing out multiple combinations. Then only move two cubes for as many shapes as possible without repeating and so on.

3.         Why is it important to sketch your ideas on paper and sign and date the document?

Sketching ideas out can help a designer visualize and see if the idea will work or not. The sign and date is important to prevent credentials from being stolen by others.

Activity 4.1b Graphical Modeling

Work

3D Modeling

Feedback: The black model drawn was kinda hard to see. They look nice with color and is overall easy to follow and assemble the pieces together.

1.     Why is it important to have designs and drawings reviewed by peers?
Peer review is essential to receive feedback and to make sure mistakes are kept to bare minimum. Moreover an extra set of eyes may see differently due to different perspectives.

Activity 4.1c Mathematical Modeling

Color of Piece	Number of Cubes	Mass (g)
Black	6	30g
Red	6	31g
Blue	5	24g
Green	4	20g
Wood	6	29g

3.         Consider the data that you have graphed and answer the following.

a.         Would you expect that this data is linear; that is, if you were to measure the mass of other pieces with more than six cubes or fewer than four cubes, would the points fall on a straight line on the graph? Explain your answer.

The data should be near linear due to cubes being manufactured at almost consistent mass. However the adding of glue varies creating slight changes in mass. 

b.         If you were to sketch a line-of-best fit, what would be a reasonable y-intercept. That is, where would the line-of-best fit cross the vertical axis? Explain your answer.

 The y intercept should be 0 grams due to nothing being placed on the electronic scale in the first place.

c.         Based on your data, what would you predict for the mass of a single wooden cube? Explain your answer or show your work.

Five grams by predicting using the line of best fit drawn on the graph.

5.         Using your line-of-best-fit, complete the following.

a.         Estimate the slope of your line-of-best-fit (include the appropriate units). Explain the interpretation of the slope in words.
The slope should be 5. When you cross over 6, the number rises to 30.

b.         Write an equation for your line of best fit.

y=5x

c.         Rewrite your equation for the line-of-best-fit in function notation where M(n) = mass of the puzzle piece and n = the number of wooden cubes.

M(n)=5n

d.         Estimate the mass of a puzzle piece that includes two wooden cubes. Show your work.

M(2)=5(2)
M(2)=10 g

e.         If a puzzle piece has a mass of 31.5 grams, how many wooden cubes would you predict were used to create the puzzle piece? Show your work.

31.5g=5n
n=6.3 cubes

7.      Use the equation of the trend line to answer the following.

a.      Rewrite the equation of the trend line using function notation where M(n) represents mass and n represents the number of cubes.

 M(n)= 4.9664n

b.      What is the domain of the function? That is, what values of n make sense?

 All real numbers above 0.

c.      What is the range of the function?

All real numbers above 0. 

d.      What is the slope of your trend line? Explain the interpretation of the slope in words.

 For every one cube added to the puzzle piece, the mass increases by 4.9664 g.

e.      How does the slope of your line-of-best-fit compare to the slope of the trend line? Why is there a difference?

 The slope of trend line is more accurate compared to estimation by line of best fit due to trend line being done by computation. While the line of best fit is done by estimation with eye.

f.        Predict the mass of a puzzle piece that is comprised of 9 wooden cubes. Show your work.

  M(9)= 4.9664(9)

M(9)=44.6976g

g.      If a puzzle piece had a mass of 49 grams, how many wooden cubes would it contain? Show your work.

49g= 4.9664n

n=9.8663

h.     Is the trend line a good representation of the relationship between the number of wooden cubes and the weight of the puzzle?  Justify your answer.

Yes it is a good representation, because by doing regression analysis. The r squared value is close to 1 with 96.74%.

Part 2.  Identify a mathematical model to represent the mass of puzzle pieces if larger wooden blocks are used.  Then use the mathematical model to make predictions.

1.      Assume that a puzzle cube was rebuilt using 1 in. cubes rather than ¾ in. cubes and the following masses were recorded for the pieces.

Number of cubes	Mass (g)
4	39
5	47
6	57
6	58
6	57

Complete each of the following.

a.      Create a scatterplot and find a trend line for the data using Excel. Print a copy of your worksheet that includes

·         Table of data

·         Scatterplot with properly formatted axes, axes labels and units, and an appropriate chart title

·         Trend line and its equation displayed on the scatterplot

b.      Write the equation relating number of cubes to mass in function notation. Be sure to define your variable.
y=9.5503x
y= Mass of Puzzle piece, x= number of cubes used

c.      What is the slope of the line (include units)? Explain the interpretation of the slope in words.

9.5503, this means for every puzzle cube added, the mass increases by 9.5503 g.

d.      How does the slope of this function (relating the number of 1 in. cubes to mass) compare to the slope of the function you found in number 5 (relating the number of ¾ in. cubes to mass)? Explain the difference.

4.9664 (slope relating to number of 3/4 inch cube) is less than 9.5503 (slope of one inch cubes). This is due to increased mass due to increase in size of cube.

e.      Use the function to predict the mass of a puzzle piece (using 1 in. cubes) if the piece includes 8 cubes. Show your work.

y=9.5503(8)
y=76.4024 g

f.        If a puzzle piece 95 grams, how many 1 in. cubes are most likely included in the piece? Show your work.
95g=9.5503x
x=9.9473 cubes
x~10 cubes

Part 3.  Find a mathematical model to represent the minimum jump height of a BMX bike as a function of the bike mass. Then use the mathematical model to make predictions.

2.      An engineer is redesigning a BMX bike. He is interested in how the mass of the bike affects the height that the bike reaches when the rider “gets air” or jumps the bike off of a ramp. He asked an experienced rider to test bikes of various masses and recorded the following minimum jump heights.

Bike Mass (lbm)	Minimum Jump Height (in.)
19	83.5
19.5	82.0
20	79.2
20.5	77.1
21	74.9
22	73.3
22.5	71.0
23	68.1
23.5	65.8
24	64.2

Use this data to complete each of the following.

a.      Create a scatterplot and find a trend line for the data using Excel. Print a copy of your worksheet that includes the following:

·         Table of data

·         Scatterplot with properly formatted axes, axes labels and units, and an appropriate chart title

·         Trend line and its equation displayed on the scatterplot

b.      Write the equation relating Bike Mass to Minimum Jump Height in function notation. Be sure to define your variable.

y = -3.7891x + 155.38

c.      What is the domain of the function? Explain.

Any real numbers above 0 cause the mass can't be zero or negative if there is matter in existence.

d.      What is the range of the function?

The range is between 155.38 and 0. 

e.      What is the slope of the line (include units). Is the slope positive or negative? Explain the interpretation of the slope in words.

-3.7891, for every pound increased the distance the bicycle jump go down by 3.7891 inch.

f.        If the engineer designed a bike that weighs 18 pounds, predict the minimum jump height. Give your answer in inches (to the nearest hundredth of an inch) and feet and inches (to the nearest inch). Show your work.

y= -3.7891(18) + 155.38

y=87.1762

g.      If the engineer designed a bike that weighs1 pound, predict the minimum jump height. Give your answer in inches to the nearest hundredth of an inch and feet and inches to the nearest inch. Show your work.

y= -3.7891(1) + 155.38

y=151.59 in or approximately 13 feet and 4 inches

h.     Does the predicted height for a one pound bike make sense? Is this function a good predictor for minimum jump heights at all bike masses? Explain.

The bike is really small and have few parts hence shouldn't have that for jump height. The function shouldn't be predictor for minimum jump height but for similar masses to the the masses used for data.

i.        If the minimum jump height of 89.7 inches is recorded, predict the estimated mass of the bike. Show your work.
89.7 in=-3.7891x +155.38 

x=17.3 lb

Extend Your Learning

3.      Assume that you will build your puzzle cube from 2 cm cubes of solid gold and each cube had a mass of 153 g. Address each of the following.

a.      Give a mathematical model (in function notation) that would represent the mass of a puzzle piece depending on the number of gold cubes used in the piece. Define your variables.

M(n)=153n
M= mass of puzzle piece
n= number of 2 cm gold cubes

b.      What would be the mass of a puzzle piece that is comprised of four gold cubes?
M(4)=153(4)
M(4)=612 g

c.      If gold sells for $60/g, what is the four-cube gold puzzle piece worth?
Since the mass is 612g, the worth is $36,720.

d.      How many gold cubes would you expect to be included in a puzzle piece that weighs 1071 g?
1071g=153(n)
n=7

Conclusion

1.      What is the advantage of using Excel for data analysis?

Excel allows the quick repetition of similar function for different data.

2.      What precautions should you take to make accurate predictions?

It's to try inputing numbers or thinking about certain results.

3.      What is a function? Explain why the mathematical models that you found in this activity are functions.

Functions are equations with different output in accordance with the input.

4.       Are all lines functions? Explain.
No because some line can be as simple as y=2. That doesn't allow any input leading to output effect.

Activity 4.1d and Activity 4.1.e - Software Modeling Introduction

Activity 4.1g Model Creation

a.	Which method (additive, subtractive, or a combination of additive and subtractive) did you use?  Why do you feel it was most efficient? I used combination since I could adjust to my needs.

b.Describe the method you used to model the object? Why do you feel it was most efficient?

I created overall dimensions then subtracted. It was most efficient cause only have two parts that are in need of removing rather that add many smaller parts.

c.Which method (additive, subtractive, or a combination of additive and subtractive) do you feel would be the most efficient method for creating the object?  Why?

I think it would have been more efficient using combination to create two different prism then subtract once each. Because carving out the L wasn't that necessary, therefore steps could be cut out. While adding only requires adding together many small parts.

b.	Describe the method you used to create the object. I created upper prism then lower prism.Then I subtract out for creating L for both prism.

c.	Can you think of a method to create the 3D solid model that would be more efficient?  Explain Maybe I more efficient method is to  be able to freely sketch the shape contour then add in dimensions. Therefore it's easier to draw and not much hassle cutting and adding.

Conclusion

1.      Why is it important to consider efficiency when planning your method of creation before beginning to model an object in CAD?

 This allows the designer to create his or her designs quickly, therefore can focus on other work

2.      How can the information provided in the browser of the CAD software help you compare the efficiency of two different methods of modeling the same object?

Conclusion

1.      Why is it important to model an idea before making a final prototype?

 Modeling allows  the designer to see visually whether an idea will work or not.

2.      Which assembly constraint(s) did you use to constrain the parts of the puzzle to the assembly such that it did not move? Describe each of the constraint types used and explain the degrees of freedom that are removed when each is applied between two parts. You may wish to create a sketch to help explain your description.

3.      Based on your experiences during the completion of the Puzzle Design Challenge, what is meant when someone says, “I used a design process to solve the problem at hand”? Explain your answer using the work that you completed for this project.

 We used the design process to create a puzzle that is challenging and meets the requirements. As well as test out prototype.

4.      How does the gender of the puzzle solver affect solution time? Be specific and provide evidence to support your answer.

 There wasn't testing on other gender during testing process.

5.      How does the age of the puzzle solver affect solution time?

a.      Make a specific statement related to the rate of increase or decrease of solution time with respect to age. Provide evidence that supports your statement.

As people age the solution time increases

b.      Write an equation using function notation that represents puzzle solution time in terms of age. Be sure to define your variables and identify units.

y=0.243x+8.84

c.      Predict the solution time on the first attempt of a child who is 3 years of age.  Show your work.

y=0.243(3)+8.84
y=9.569 min

d.      Predict the solution time on the first attempt of a person who is 95 years of age. Show your work.

y=0.422x+8.81
y=0.422(95)+8.81
7=48.9 min

e.      Do these predictions make sense?  Why or why not?

The prediction for older person makes sense however the prediction for young child doesn't make sense. Because a child that young will need lots of thinking and concentration.

f.        What is a realistic domain for the function?

Between 14-57 due to having subjects of those age range.

Final Document:
https://docs.google.com/document/d/1BQvr-NW_nDp_BTRBA96HjceXGqPH78s0BX6rTrZKuIg/edit?usp=sharing