Question

there are two lemonade stands that sell lemonade called stand A and stand B. The total cost for lemonade at Stand A is $15 (C) plus $2 per ice (I). The total cost for lemonade at Stand B is $17 (C) plus $4.50 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, let's express the costs using the provided formulas.

**Stand A:**
- Total Cost, \( C_A = 15 + 2I \)

**Stand B:**
- Total Cost, \( C_B = 17 + 4.5I \)

To determine which lemonade stand is cheaper, we can set up an inequality or simply compare the costs for different values of \( I \) (the number of ice units).

1. **When \( I = 0 \) (no ice):**
- \( C_A = 15 + 2(0) = 15 \)
- \( C_B = 17 + 4.5(0) = 17 \)
- Here, Stand A is cheaper.

2. **When \( I = 1 \):**
- \( C_A = 15 + 2(1) = 17 \)
- \( C_B = 17 + 4.5(1) = 21.5 \)
- Here, Stand A is still cheaper.

3. **When \( I = 2 \):**
- \( C_A = 15 + 2(2) = 19 \)
- \( C_B = 17 + 4.5(2) = 26 \)
- Here, Stand A remains cheaper.

4. **When \( I = 3 \):**
- \( C_A = 15 + 2(3) = 21 \)
- \( C_B = 17 + 4.5(3) = 30.5 \)
- Stand A continues to be cheaper.

You can see a pattern that Stand A remains cheaper than Stand B at least for these values of \( I \).

To find the point at which the costs of Stand A and Stand B are equal, we set the equations equal to each other:

\[
15 + 2I = 17 + 4.5I
\]

Solving for \( I \):
\[
15 - 17 = 4.5I - 2I
\]
\[
-2 = 2.5I
\]
\[
I = -\frac{2}{2.5} = -0.8
\]

Since the number of ice units \( I \) cannot be negative, Stand A will always be cheaper than Stand B for non-negative values of \( I \).

Thus, **Lemonade Stand A is cheaper than Stand B for all valid values of \( I \) (number of ice units).**