The Sugar Sweet Company will choose from two companies to transport its sugar to market. The first company charges $4500 to rent trucks plus an additional fee of $200.25 for each ton of sugar. The second company charges $4041 to rent trucks plus an additional fee of $225.75 for each ton of sugar.

Let's define the cost functions for both transportation companies to understand their pricing structure. We'll use the following variables:

- Let $ x $ represent the number of tons of sugar transported.

### Cost Function Analysis

1. **First Company:**
   The total cost is the sum of the fixed rental cost and the variable cost based on the tons of sugar.  
   \[
   C_1 = 4500 + 200.25x
   \]

2. **Second Company:**
   Similarly, the total cost for the second company is:
   \[
   C_2 = 4041 + 225.75x
   \]

### Determining the Better Option
To find the number of tons $ x $ for which each company is more cost-effective, we can set $ C_1 $ equal to $ C_2 $ and solve for $ x $.

\[
4500 + 200.25x = 4041 + 225.75x
\]

Simplifying:

\[
4500 - 4041 = 225.75x - 200.25x
\]
\[
459 = 25.5x
\]
\[
x = \frac{459}{25.5} \approx 18
\]

### Interpretation
- When $ x = 18 $ tons, both companies will have the same cost.
- For less than 18 tons, the second company (lower fixed cost) will be cheaper.
- For more than 18 tons, the first company (lower variable cost) will be cheaper.

Would you like to see the detailed calculations for specific tonnages, or do you have any further questions?

### Related Questions:
1. What will be the total cost for each company if 10 tons of sugar are transported?
2. At what point will the cost difference between the two companies be most significant?
3. How much will Sugar Sweet Company save by choosing the more cost-effective option for 30 tons?
4. If the variable costs change, how will that affect the break-even point?
5. What if the fixed rental costs were reduced for both companies?

**Tip:** Always check the break-even point when comparing linear cost functions to determine which option is more economical based on quantity.

Learn how to compare costs for transporting sugar using two companies with different rental and per-ton charges. This detailed analysis shows when each company becomes more cost-effective based on tonnage. Includes step-by-step solution and break-even point calculation.

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