small grocery store offers two different brands of milk.

Question

Brand A 2% milk costs $3.50 a gallon.
Brand B whole organic milk costs $3.00 for a half-gallon plus a one-time deposit of $1.50 for the glass jug. Once the glass jug deposit has been paid, the jug can be brought back to the store and the deposit doesn't have to be paid for each additional half-gallon milk that is purchased.
Graph the cost of milk for the brand that is proportional.

Answer 1

To graph the cost of the two brands of milk, let's first establish the costs for both Brand A and Brand B.

Brand A

• Cost per gallon of 2% milk: $3.50

Since this is proportional and we are considering price per gallon, we can express the cost \( C_A \) of Brand A as: \[ C_A = 3.50x \] Where \( x \) is the number of gallons.

Brand B

• Cost per half-gallon of whole organic milk: $3.00
• One-time deposit for glass jug: $1.50

For Brand B, we will have two scenarios:

1. The initial purchase including the deposit: \[ C_B = 1.50 + 3.00y \] where \( y \) is the number of half-gallons purchased. The first half-gallon will include the deposit.

2. For subsequent purchases (after we already paid the deposit for the jug): \[ C_B = 3.00y \] This equation applies when \( y \) (number of half-gallons) is greater than or equal to 1.

To convert the cost of Brand B into a per-gallon basis for easier graphing:

• 1 gallon = 2 half-gallons.

So the cost function for Brand B can be expressed as:

• For the first half-gallon (with deposit): \[ C_B(1/2) = 1.50 + 3.00 \cdot 1 = 4.50 \] (équals about $9.00 per gallon for the first half-gallon)

• For any additional half-gallons (after the deposit): \[ C_B(y/2) = 3.00y \quad \text{For } y \geq 1 \] And we convert this back to a per gallon basis: \[ C_B = 6.00z \quad \text{where } z \text{ is the number of gallons} \]

Graphing the Costs

Brand A:

• Straight line starting at (0, 0) with a slope of 3.50 (cost per gallon).

Brand B:

• Starts at (0, 1.50) for the deposit and first half-gallon, \( (0.5, 4.50) \) for the first half-gallon total.
• After \( 1 \) gallon, the cost is linear with a slope of 6.00 (this means every half gallon adds $3.00 continuously after the first).

Steps to Graph

1. Plot Brand A:

   • Point at (0, 0) and (1, 3.50)
   • Continue this line with slope of 3.50.

2. Plot Brand B:

   • Start at (0, 1.50) for the deposit.
   • For (0.5, 4.50) and (1, 7.50).
   • After that use a slope of $6.00 per gallon from the second half-gallon on.

Visual Overview

The final graph will show Brand A as a straight line beginning at $0 going up to $3.50 per gallon, while Brand B will show a step increase initially due to the deposit and then rise quicker after the first gallon as a steeper line indicating higher costs.

Make sure to label the axes with "Quantity (Gallons)" on the x-axis and "Cost ($)" on the y-axis, and clearly distinguish between the two brands of milk.