1. A cab ride costs $10 per mile. Change this scenario to non proportional.

Question

2.A cab ride costs $10 to get into the cab and then $4 per mile. Change this to proportional.

3.You want to sell t-shirts for $12 a piece. It cost you $40 for supplies. Is the total profit proportional to the number of t-shirts you sell?

4. Change the scenario in #3 to be a proportional scenario.

5.Show a proportional graph and equation and compare to a non proportional graph and equation.
Proportional: y =



Non Proportional: y =

Answer 1

Let's go through your questions one by one:

1. Change the cab ride cost scenario to non-proportional.

Original: A cab ride costs $10 per mile.
Non-Proportional Change: A cab ride costs $10 for the first mile and $10 for each additional mile after that. (Now, the first mile has a fixed cost that doesn't rely on the variable rate.)

2. Change the cab ride scenario to proportional.

Original: A cab ride costs $10 to get into the cab and then $4 per mile.
Proportional Change: A cab ride costs $4 per mile with no initial fee (meaning the total cost is directly proportional to the number of miles driven). For example, if you drive 1 mile, you pay $4, for 2 miles you pay $8, etc.

3. Is the total profit from selling t-shirts proportional to the number of t-shirts you sell?

• Selling price per t-shirt: $12
• Cost for supplies: $40
• To find the profit per shirt, we need to consider:
  • Profit per t-shirt: \( 12 - \text{cost per t-shirt} \)
  • Let's say it costs you \( \frac{40}{x} \) (where \( x \) is the number of shirts). Thus:
  • Profit = \( 12x - 40 \)
• This shows that the total profit is not proportional to the number of t-shirts sold because the fixed cost of $40 means that the relationship includes a constant term.

4. Change the scenario to be proportional.

Proportional Change: Suppose you only pay a fixed cost of $2 per t-shirt (instead of $40 upfront). If you sell t-shirts for $12 each, the profit can be expressed as:

• Profit = \( 12x - 2x = 10x \)
• The total profit is now proportional to the number of t-shirts sold.

5. Proportional vs Non-Proportional Graph and Equation

Proportional Relationship:

• Equation: \( y = 4x \) (where \( y \) is cost and \( x \) is miles for the proportional cab ride scenario)
• Graph: A straight line through the origin (0,0) with a slope of 4.

Non-Proportional Relationship:

• Equation: \( y = 10 + 10(x - 1) = 10x \) (for the non-proportional cab ride scenario, representing a fixed cost for the first mile)
• Graph: A straight line that starts at the point (0,10) and then has a slope of 10, indicating a fixed cost at the beginning.

Summary:

• Proportional:

  • Equation: \( y = 4x \)
  • Graph: Line through the origin with a positive slope.

• Non-Proportional:

  • Equation: \( y = 10 + 10x - 10 \Rightarrow y = 10x \)
  • Graph: Line that intersects the y-axis at the fixed cost, not through the origin, indicating a starting cost.