Question

there are two lemonade stands that sell lemonade called stand A and stand B. The total cost for lemonade at Stand A is $8 (C) plus $2 per ice (I). The total cost for lemonade at Stand B is $7 (C) plus $2.50 per ice (I). Which Lemonade stand is cheaper. Lets use C for the total cost and I for the cost per ice.

Answer 1

To compare the total costs of lemonade at Stand A and Stand B, we can express the total cost for each stand as a function of the number of ice cubes (I).

**Stand A:**
- Total cost \( C_A = 8 + 2I \)

**Stand B:**
- Total cost \( C_B = 7 + 2.5I \)

To determine which stand is cheaper, we can set up an inequality:

1. **Set the costs equal to find the break-even point:**
\[
8 + 2I = 7 + 2.5I
\]

2. **Solve for I:**
\[
8 - 7 = 2.5I - 2I
\]
\[
1 = 0.5I
\]
\[
I = 2
\]

This means that when \( I = 2 \) (2 ice cubes), the total costs of both stands will be the same.

Now we can find the total costs for both stands when \( I < 2 \) and \( I > 2 \):

- **For \( I < 2 \):**
- Stand A: \( C_A = 8 + 2I \)
- Stand B: \( C_B = 7 + 2.5I \)

Let's check at \( I = 0 \):
- \( C_A = 8 + 2(0) = 8 \)
- \( C_B = 7 + 2.5(0) = 7 \)

Here, **Stand B is cheaper.**

- **For \( I > 2 \):**
- Stand A: \( C_A = 8 + 2I \)
- Stand B: \( C_B = 7 + 2.5I \)

Let's check at \( I = 3 \):
- \( C_A = 8 + 2(3) = 14 \)
- \( C_B = 7 + 2.5(3) = 12.5 \)

Here, **Stand A is cheaper.**

### Conclusion:
- Stand B is cheaper when the number of ice cubes is less than 2.
- Stand A is cheaper when the number of ice cubes is greater than 2.
- At exactly 2 ice cubes, the total costs are the same.