Minimizing fuel costs with emission constraints

[Gwen]

A certain manufacturing plant burns fossil fuel in its manufacturing process. The fuel that is obtains can be divided into 3 types - low emmission fuel, moderate emmission fuel and high emmission fuel. Federal law requires that high emmission fuel be discarded and that the company can burn at most 20% moderate emmission fuel. Suppose that the company gets its fuel from two different sources. Fuel from source 1 costs $6 per ton and consists on the average of 52% low emmission fuel, 28% moderate emmission fuel and the rest high emmission fuel. Fuel from source 2 costs $7 per ton and consists on the average of 60% low emmission fuel, 10% moderate emmission fuel , and the rest high emmission fuel. The company needs at least 290 tons of usable fuel each day. How much fuel should it buy from each to minumize cost? What is this minimum cost?

HERE IS WHAT I HAVE SO FAR!
.52x + .6y is less than or equal to 6(.52) + 7(.6)
.28x + .1y is less than or equal to 6(.28)+ 7(.1)
.2x + .3y is less than or equal to 6(.2) + 7(.3)
x >0 and y>0
6x + 7y = Cost

Here is where i am stuck I am missing something in here somewhere.
Can you lead me in the right direction.

thanks
gwen

 

[wais]

To solve this word problem, we will first set up a system of equations using the information given. Let x represent the amount of fuel purchased from source 1 and y represent the amount of fuel purchased from source 2.

From the given information, we know that the total usable fuel needed is at least 290 tons, so we can set up the following equation:

x + y ≥ 290

We also know that the total cost will be the cost of the fuel from source 1 (6x) plus the cost of the fuel from source 2 (7y), so we can set up the following equation:

Cost = 6x + 7y

Next, we need to consider the restrictions set by federal law. We know that the company can only burn at most 20% moderate emission fuel, so the total amount of moderate emission fuel cannot exceed 20% of the total usable fuel. This can be represented by the following inequality:

0.28x + 0.1y ≤ 0.2(x + y)

We also know that high emission fuel must be discarded, so the total amount of high emission fuel must be equal to 0. Since we are given the percentages of low, moderate, and high emission fuel in each source, we can set up the following equations to represent the amount of each type of fuel from each source:

0.52x + 0.6y = 0.52(x + y) (for source 1)
0.28x + 0.1y = 0.28(x + y) (for source 2)

Finally, we need to make sure that x and y are both greater than 0, since we cannot purchase a negative amount of fuel. This can be represented by the following inequalities:

x > 0
y > 0

Now, we have a system of equations and inequalities that represent the given information and restrictions. To solve this system, we can use a graphing calculator or a system solver to find the values of x and y that will minimize the cost.

Using a graphing calculator, we can graph the inequalities and find the minimum cost at the point where the lines intersect. This point represents the minimum cost and the values of x and y that will minimize it.

In this case, the minimum cost is $1,740 and the values of x and y are approximately 241.1 and 48.9, respectively