The Sugar Sweet Company will choose from two companies to transport its sugar to market. The first company charges 6443 to rent trucks plus an additional fee of 175.75 for each ton of sugar. The second company charges 7500 to rent trucks plus an additional fee of 100.25 for each ton of sugar. What is the cost when the two companies charge the same?

Let's define the cost functions for both companies based on the information provided:

1. **First Company**:
   - Fixed truck rental cost: \$6,443
   - Cost per ton of sugar: \$175.75
   - Total cost for the first company:  
   \[
   C_1 = 6443 + 175.75x
   \]
   where $x$ is the number of tons of sugar transported.

2. **Second Company**:
   - Fixed truck rental cost: \$7,500
   - Cost per ton of sugar: \$100.25
   - Total cost for the second company:  
   \[
   C_2 = 7500 + 100.25x
   \]

We need to find when the total cost for both companies is the same, i.e., $C_1 = C_2$. Setting the equations equal to each other:

\[
6443 + 175.75x = 7500 + 100.25x
\]

Now, let's solve this equation.

1. Subtract $100.25x$ from both sides:
   \[
   6443 + 75.5x = 7500
   \]

2. Subtract 6443 from both sides:
   \[
   75.5x = 1057
   \]

3. Divide both sides by 75.5 to solve for $x$:
   \[
   x = \frac{1057}{75.5} \approx 14
   \]

Thus, the two companies will charge the same when approximately **14 tons** of sugar are transported.

Would you like further details on the solution or have any questions?

Here are 5 related questions to explore further:
1. How does the cost comparison change if the price per ton increases for both companies?
2. What if the rental fee of the second company decreases by 10%?
3. How many tons must be transported for the first company to always be cheaper?
4. What would happen if both companies added an additional fixed surcharge on top of the rental cost?
5. How does increasing or decreasing the tonnage impact the break-even point between the two companies?

**Tip**: When comparing costs over variable quantities, setting up cost functions and equating them is an effective way to find break-even points.

Solve a cost comparison problem between two companies transporting sugar to market. Learn how to set up and solve linear equations to find the break-even point where both companies charge the same. Suitable for students in Grades 7-10.

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