Let's Beat the French: Create a Game-Changing Drink Bottle for Brisbane 2032!

Question

The Brisbane City Council is looking to showcase Australian creativity and heritage by marketing a new water bottlereo celebrate the upcoming Brisbane 2032 Olympics. They have tasked you ing designing a 1-litre bottle that not only serves its practical purpose but also harporales a logo that represents all Australians and honours our history and Aboriginal heritage. Your ear 9 mathematics class has been enlisted to present a comprehensive repare. some design proposals, which the Council can then use to create a proposal for their manume desto reduce material use and promote sustainability.

Design Brief:
1. Variety in Design:
- You are required to design two different containers.
- The shape of the cartons should be selected from the following groups: prisms, pyramids, cylinders, or cones.
2. Volume and Shape:
- The volume of each package must be 1 litre (1000cm*), but the shapes must differ from the standard cylindrical drink containers.
3. Minimizing Material Use:
- The surface area (amount of material used) of the container should be calculated and kept to a minimum.
4. Cultural Representation:
- Each design must feature a logo that creatively represents all Australians and acknowledges our rich history and Aboriginal heritage.

Submission Requirements:
your proposal to the brisbane city counsil must include the following components:
1. Planning document:
- a detailed planning document outlining your designs.
- A recommendation for which design the council should use, supported by reasons. Consideration should include the amount of material used and practical issues that supermarkets may encounter, such as staking and shelf space.
2. 3-D Sketches:
- clear, 3-dimensional sketches of your proposed designs.
3. Scale Drawings:
- Neat, to-scale drawings of the nets for your cartoons, labelled with dimensions.
4. Calculations:
- detailed calculations demonstrating that the volume of the containers meets the 1-litre requirement.
- Calculations of the surface area for each design, highlighting the martial usage.
5. Logo Design:
- A creative logo that represents all Australians and our history, with a focus on Aboriginal heritage.
6. Model:
- An “actual size” model of your recommended design for a bottle.

PLANNING DOCUMENT
1
Interpret the task.

Write a short description of the task.
Designing a new 1-litre drink water bottle for the Brisbane’s 2023 olympics. The bottle design unique, minimising martial use while representing Australian heritage, particularly focusing on Aboriginal culture. We are required to create two different container shapes, calculate their volumes and surface areas, and provide visual and practical proposals for the councils consideration.

List the information provided to help you complete the task?
- The drink bottle must hold a volume of 1 litre (1000 cm^3).
- The shapes must differ from standard cylindrical containers.
- The surface are should be minimised.
- Each design must feature a logo reflecting Australian and Aboriginal heritage

List other information you may need to find out to complete the task fully.
- The appropriate dimensions for the selected shapes to achieve the required volume.
- The surface area formulas for different geometric shapes.
- Information on materials that can be used for sustainable practices.

2
Identify the Mathematics

List the units of measurement you will need to consider.
- Volume = cubic centimetres (cm^3)
- Surface Area = Squared centimetres (cm^2)
- Dimensions = Centimetres (cm)

List any formula that might be useful to complete this task.
- Volume of rectangular prism = Length * Width * Height
- Volume of pyramid = 1/3 * Base area * Height
- Volume of a cone = 1/3 * pi r^2 Height ( r= radius)
- Volume of a cylinder = pi r^2 Height
- Surface area of a rectangular prism = 2(LW+LH+WH)
- Surface area of a pyramid = Base + 1/2 * perimeter of the base * length
- Surface area of a cone = pi r(r+L) (R=radius, L=length)
3
Analyse information and data sources.

List where or who you plan to get the additional information from.
- Research on Australian Aboriginal culture and symbols to create a meaningful logo.
- My exercise book
- Khan academy

4
Applying the mathematics

Write down any calculations you use to work out the dimensions of your design.

Calculate the volume:

Calculate the surface area:

DESIGN 1

write down any calculations you use to work out the dimensions of your design.

Draw a labelled sketch of your first container design. Include a brief description of the 3D shape/s you are using.

Draw a scale drawing of the net for design 1 and calculate the surface area:

DESIGN 2
write down any calculations you use to work out the dimensions of your design

Draw a labelled sketch of your second container design. Include a brief description of the 3D shape/s you are using.

Draw a scale drawing of the net for design 2 and calculate the surface area:

5
State and Verify Solution.

State your preferred model.

Answer 1

PLANNING DOCUMENT

1. Interpret the Task

Short Description of the Task: We are tasked with designing a new 1-litre drink bottle for the Brisbane 2032 Olympics. The bottle design should be unique, minimize material use, and represent Australian heritage, particularly focusing on Aboriginal culture. We will create two different container shapes, calculate their volumes and surface areas, and provide visual and practical proposals for the council's consideration.

List of Information Provided:

• The drink bottle must hold a volume of 1 litre (1000 cm³).
• The shapes must differ from standard cylindrical containers.
• The surface area should be minimized.
• Each design must feature a logo reflecting Australian and Aboriginal heritage.

Additional Information Needed:

• Appropriate dimensions for the selected shapes to achieve the required volume.
• Surface area formulas for different geometric shapes.
• Information on sustainable materials that can be used.

2. Identify the Mathematics

Units of Measurement:

• Volume = cubic centimeters (cm³)
• Surface Area = square centimeters (cm²)
• Dimensions = centimeters (cm)

Useful Formulas:

• Volume of a rectangular prism \( V = L \times W \times H \)
• Volume of a pyramid \( V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \)
• Volume of a cone \( V = \frac{1}{3} \pi r^2 h \)
• Volume of a cylinder \( V = \pi r^2 h \)
• Surface area of a rectangular prism \( SA = 2(LW + LH + WH) \)
• Surface area of a pyramid \( SA = \text{Base} + \frac{1}{2} \times \text{Perimeter of Base} \times \text{Slant Height} \)
• Surface area of a cone \( SA = \pi r(r + L) \)

3. Analyze Information and Data Sources

Information Sources:

• Research on Australian Aboriginal culture and symbols to create a meaningful logo.
• My textbooks and exercise books for mathematical formulas and problem-solving strategies.
• Khan Academy and other online educational platforms for geometric principles.
• Material science resources for sustainable practices.

4. Applying the Mathematics

Calculating Dimensions and Volumes:

DESIGN 1: Rectangular Prism

Dimensions: Let's assume a base dimension of \( L = 10 , cm \) and \( W = 10 , cm \), we can find the height \( H \) using the volume formula.

1. Volume = \( L \times W \times H \)
   • \( 1000 cm³ = 10 cm \times 10 cm \times H \)
   • \( H = \frac{1000 cm³}{100 cm²} = 10 cm \)

Surface Area Calculation:

• Surface Area = \( 2(LW + LH + WH) \)
• Surface Area = \( 2(1010 + 1010 + 10*10) \)
• Surface Area = \( 2(100 + 100 + 100) = 600 cm² \)

Sketch of Design 1:

Provide a labeled sketch of the rectangular prism here. Highlight the dimensions (10 cm x 10 cm x 10 cm).

Scale Drawing of the Net for Design 1: Provide a neat, to-scale drawing with labels for each face (front, back, sides, top, bottom).

DESIGN 2: Pyramid

Dimensions: Assuming a square base with side length \( a \).

1. Volume = \( \frac{1}{3} \times a^2 \times H \)
   • If we let \( a = 10 , cm \),
   • Then \( H = \frac{1000 cm³}{\frac{1}{3} \times 10^2} = \frac{1000 cm³}{\frac{1}{3} \times 100} = 30 , cm \).

Surface Area Calculation:

• Surface Area of a pyramid (1 base + 4 triangular faces).

Base area = \( a^2 = 100 cm² \) and slant height \( \ell \) can be calculated using Pythagorean theorem:

• \( \ell = \sqrt{(\frac{a}{2})^2 + H^2} = \sqrt{(5)^2 + (30)^2} = \sqrt{25 + 900} = 31.62 cm \)

Surface Area = Base + 4 triangular faces

• Surface Area = \( 100 + 4 \times \frac{1}{2} \times a \times \ell \)
• Surface Area = \( 100 + 2 \times 10 \times 31.62 = 100 + 632.4 = 732.4 cm² \)

Sketch of Design 2:

Provide a labeled sketch of the pyramid with dimensions (base 10 cm x 10 cm, height 30 cm).

Scale Drawing of the Net for Design 2: Provide a neat, to-scale drawing of the pyramid net, clearly labeling the slant heights and base dimensions.

5. State and Verify Solution

Preferred Model: I recommend the Rectangular Prism Design due to:

• Lower surface area (600 cm² compared to 732.4 cm² of the pyramid).
• Simpler stacking in retail environments like supermarkets.
• Easier manufacturing and potential for better recyclability with the materials selected.

Logo Design:

• Create a logo featuring elements that symbolize Australia and Aboriginal culture, possibly incorporating the Southern Cross constellation and traditional Aboriginal patterns.

Model:

• Construct an “actual size” model of the rectangular prism design for presentation, showcasing material use and practical storage solutions.

This comprehensive proposal should align well with the Brisbane City Council's goals and vision for the 2032 Olympics water bottle design.