Use the following situation for all parts of the question.

Question

To raise money for a field trip, the student council is selling popcorn. It costs them $0.50 per bag plus $6 for signage. They are selling the popcorn for $2.00 per bag.

Part 1:

Given y is the total cost and x is the number of bags, identify the correct equations to represent this situation.

Responses

y = x + 2

y = 6x + 0.5

y = 0.5x + 6

y = 2x

Answer 1

To analyze the situation and represent it with the correct equations, we need to look at the costs and revenues involved in selling the popcorn.

1. Total Cost (y): The total cost includes a fixed cost for signage and a variable cost that depends on the number of bags. The cost for the bags is $0.50 per bag, and the fixed signage cost is $6. Therefore, the equation representing total cost can be expressed as:

   \( y = 0.5x + 6 \)

2. Revenue (not explicitly required in this part but useful for understanding): The revenue from selling the popcorn is determined by the number of bags sold at $2.00 per bag. This revenue equation would be:

   \( y = 2x \)

However, since the prompt specifically asks for the equation representing the total cost, the correct choice is:

y = 0.5x + 6

Answer 2

Two answers are correct here in this case.

Answer 3

Thank you for your clarification. In this situation, the two equations that can be considered correct based on the context provided are:

1. Total Cost Equation:

   • \( y = 0.5x + 6 \) (This represents the total cost, as it includes the fixed cost for signage and the variable cost per bag.)

2. Revenue Equation:

   • \( y = 2x \) (This represents the total revenue from selling the popcorn bags.)

Both equations provide meaningful information in the context of the problem. "Total cost" relates to the expenses incurred by the student council, while "revenue" pertains to the income generated from sales. Thank you for pointing that out!

Answer 4

Part 3:

How many bags of popcorn must they sell to for the cost and profit to be equal?

Part 4:

What is the dollar value where the cost and profit are equal to each other?

Answer 5

To find the point where the cost and profit are equal, we first need to understand the equations for cost, revenue, and profit.

Part 3:

1. Define the equations.

• Total Cost (C): \( C = 0.5x + 6 \)
• Total Revenue (R): \( R = 2x \)
• Profit (P), which is Revenue minus Cost: \( P = R - C = 2x - (0.5x + 6) = 2x - 0.5x - 6 = 1.5x - 6 \)

2. Set Cost equal to Profit.
We need to find the number of bags sold (x) when total cost is equal to profit:

\[ 0.5x + 6 = 1.5x - 6 \]

3. Solve for x.

• Rearrange the equation:

  \[ 6 + 6 = 1.5x - 0.5x \]

  \[ 12 = 1.0x \]

• Therefore:

  \[ x = 12 \]

Part 4:

Find the dollar value for the cost and profit when they are equal.
To find the dollar value at which cost and profit are equal, we can plug \( x = 12 \) back into either the cost or profit equation.

1. Calculate Total Cost (C) when \( x = 12 \):

   \[ C = 0.5(12) + 6 = 6 + 6 = 12 \]

2. Calculate Total Profit (P) when \( x = 12 \):

   \[ P = R - C = 2(12) - (0.5(12) + 6) \] \[ P = 24 - 12 = 12 \]

Thus, both the cost and profit equal $12 when \( x = 12 \) bags are sold.

Summary:

• Part 3: They must sell 12 bags of popcorn for cost and profit to be equal.
• Part 4: The dollar value where cost and profit are equal is $12.