You have worked with figures that are similar and have learned how to calculate angles and side lengths to identify similarity. Put these skills together to plan a design for a scale model. Showing how these concepts effectively work in the real world involves clear communication and the ability to defend your conclusions to others.

Question

using a cell phone as the real-world figure

In your post, describe your design to your peers:

State the real-world figure you will model and the scale factor you will use.

Identify at least two different but significant items on your model.

Explain how you will determine the calculations and dimensions for these items to make them mathematically similar to the actual items.
Describe the specific mathematical skills you will need to create the design and give examples.
Describe some outside sources you may need for the calculations.

Answer 1

Design Plan for a Scale Model of a Cell Phone

Real-World Figure

The real-world figure I will model is a smartphone, specifically the latest iPhone model, which has precise dimensions that are widely known. For this model, I will use a scale factor of 1:4, meaning every length on the scale model will be one-fourth of the actual dimensions of the iPhone.

Significant Items on the Model

1. Screen: The screen is a crucial feature of the smartphone, with dimensions approximately 6.1 inches in diagonal length for the iPhone 13.

2. Camera Module: The camera module on the back of the phone is another significant item, with dimensions around 2.5 inches in width and 1.5 inches in height.

Calculating Dimensions for Similarity

To ensure that the scale model is mathematically similar to the actual smartphone, I will perform the following calculations for the identified items:

1. Screen Calculation:

   • Actual Diagonal Length: 6.1 inches.
   • Scale Model Diagonal Length: \( \frac{6.1 \text{ inches}}{4} = 1.525 \text{ inches} \).
   • To find the width and height, I will research the aspect ratio of the actual screen (which is approximately 19.5:9). Using this ratio:
     • Let the width be \( w \) and the height be \( h \).
     • The aspect ratio gives us the equation: \( w:h = 19.5:9 \).
     • With the diagonal calculation, I can derive \( w \) and \( h \) using the Pythagorean theorem:
       • \( (1.525)^2 = w^2 + h^2 \).

2. Camera Module Calculation:

   • Actual Width: 2.5 inches and Height: 1.5 inches.
   • Scale Model Width: \( \frac{2.5 \text{ inches}}{4} = 0.625 \text{ inches} \).
   • Scale Model Height: \( \frac{1.5 \text{ inches}}{4} = 0.375 \text{ inches} \).

These calculations ensure that the scale model maintains proportions and dimensions that are similar to the actual smartphone.

Mathematical Skills Required

To create this design, I will need the following mathematical skills and knowledge:

• Scale Factors: Understanding and applying scale factors to diminish real dimensions into model dimensions.
• Ratios and Proportions: Leveraging the aspect ratio to find the height and width of the screen.
• Pythagorean Theorem: Applying the Pythagorean theorem to calculate dimensions based on the diagonal measurement of the screen.
• Algebraic Manipulation: Rearranging equations to solve for unknowns in terms of known dimensions.

Outside Sources for Calculations

To facilitate accurate calculations and dimensions, I may need to refer to:

• Manufacturer Specifications: These would provide me with accurate dimensions for the smartphone I'm modeling.
• Online Dimension Converters: Tools that can help in conversion to ensure I apply the correct scale factor.
• Mathematics Textbooks or Websites: These can provide examples and explanations of scale models, ratios, and the Pythagorean theorem.

In conclusion, creating a scale model of a smartphone using a 1:4 scale factor involves understanding and applying geometric principles to ensure that the model is mathematically similar, effectively replicating both the screen and camera module dimensions.