Question

the sugar sweet company will choose from two companies to transport its sugar to market. the first company charges $4207 to rent trucks plus an additional fee of $15.75 for each ton of sugar. the second company charges $5500 to rent trucks plus an additional fee of $100.25 for each ton of sugar.
for what amount of sugar do the two companies charge the same?
what is the cost when the two companies charge the same?

Explanation:

Step1: Define variables and cost functions

Let \( x \) be the number of tons of sugar. Let \( C_1(x) \) be the cost function for the first company and \( C_2(x) \) be the cost function for the second company.
For the first company: \( C_1(x)=4202 + 315.75x\)
For the second company: \( C_2(x)=5500 + 100.25x\)

Step2: Set cost functions equal

To find when the two companies charge the same, set \( C_1(x)=C_2(x) \):
\( 4202 + 315.75x=5500 + 100.25x\)

Step3: Solve for \( x \)

Subtract \( 100.25x \) from both sides:
\( 4202 + 315.75x- 100.25x=5500 + 100.25x-100.25x\)
\( 4202 + 215.5x=5500\)

Subtract 4202 from both sides:
\( 215.5x=5500 - 4202\)
\( 215.5x=1298\)

Divide both sides by 215.5:
\( x=\frac{1298}{215.5}\)
\( x = 6\)

Answer:

The two companies charge the same when the amount of sugar is 6 tons.