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Question

The Sugar Sweet Company will choose from two companies to transport its sugar to market. The first company charges $6197 to rent trucks plus an additional fee of $225.75 for each ton of sugar. The second company charges $6500 to rent trucks plus an additional fee of $200.50 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?

Answer 1

This is a 3 steps problem:

1) you have to determine the expression of the total charge for company 1 as a funtion of the amount of sugar being tonned. So we’re going to represent the amount of sugar tonned by the company as x and the total charge of company 1 as f(x) because the charge is dependent upon the amount tonned and this indicate that we have a function.

expression for company 1: f(x) = 225.75x + 6197

because the 6197$ represents the fixed fee charged for renting the truck. The question added that company will charge 225.75$ for one ton of sugar therefore for a certian amount x of sugar the company will charge 225.75x$ plus the fixed fee.

2) the expression of the total charge of company 2:

let g(x) represent the the total charge of company 2.

The expression will be in that case:

g(x)= 200.50x + 6500 for the same reasons that I explained above For the expression of company 1.

3) finally set this expressions equal and solve for x:

200.50x + 6500 = 225.75x + 6197

now we should isolate x by making the right side for x and the left side for the constant numbers. In order to do that subtract 200.50x from both sides and subtract 6197 from both sides:

200.50x - 200.50x + 6500 - 6197 = 225.75x - 200.50x [phone]

this will leave us with the following:

303 = 25.25x Now divide both side by 25.25 to get x

x= 303/ 25.25

x= 12

therefore the amount of sugar for which the two companies charge the same is 12

In order to calculate the charge you can either plug in x in f(x) or g(x) it will give you the same answer because at x=12, f(x)= g(x)

you can use g(x) because it’s easier in calculation:

g (12) = 200.50 (12) + 6500 = 8906$

Answer 2

Let t represent the tonnage to be shipped.

The cost of using the first company is:

C₁(t) = 6197 + 225.75t

The cost of using the second company is:

C₂(t) = 6500 + 200.50t

We seek the value of t when C₁(t) = C₂(t), or equivalently, when C₁(t) - C₂(t) = 0.

Subtract the two equations:

C₁(t) - C₂(t) = -303 + 25.25t

Thus, we solve for t in:

0 = -303 + 25.25t

t = 303 / 25.25 = 3(101) / [(0.25)(101)] = 3 / 0.25 = 3*4

t = 12, ans

At this value of t, C₁(t) = C₁(12) = 6197 + 225.75(12)

C₁(12) = 8906 tons, ans

Check:

C₂(t) = 6500 + 200.50t

C₂(12) = 6500 + 200.50(12) = 8906.