Question 419279: COuld somene please help me solve this:
A manufacturer of synfuel ( synthetic fuel from coal) estimates that the cost c in dollars per barrel for a production run of x thousand barrels is given by c(x) = 9 x² - 180 x + 940.
How many thousands of barrels should be produced during each run to minimize the cost per barrel, and what is the minimum cost per barrel?
Answer by duckness73(47)   (Show Source):
You can put this solution on YOUR website! If this question is for a calculus course, then you can take the first derivative of the equation, set it to zero, solve for x, and you will get the number of thousands of barrels for which the cost (c(x)) is minimized. If you are not a calculus student, a different approach is needed:
Note that this equation is for a parabola. Because the leading coefficient (9) is positive, the parabola is opening up. Also, since there is no solution for  = 0 (which you should verify using the quadratic formula), the parabola never crosses the x-axis. Therefore, the vertex of the parabola is either in the first or second quadrant.
Thinking about the "real world" problem the manufacturer of the synfuel is looking for the value of x for which c(x) (the y-coordinate of the parabola)is the smallest. This means that you are looking for the lowest point in the graph, which is the vertex. We know that the vertex is either in the first or second quadrant, and since the "real world" problem doesn't make sense if the number of barrels (x) is negative, the vertex must be in the first quadrant.
So how do you find the coordinates of the vertex of a parabola? We know that the parabola is symmetric about the vertex, so each point of the parabola has a "reflection" on the other side of the vertex. So, let's just pick any value for x, find the corresponding value for y, find the other value for x with the same value of y, and calculate the mid-point between the two x's.
The easiest x to pick is x = 0, then c(0) = 940. What other value for x is c(x) = 940?
c(x) = = 940
 = 0 (subtracting 940 from both sides)
9x(x - 20) = 0 (factoring)
x = 0 (which we already knew) or x = 20
The vertex of the parabola will have an x-coordinate half way between 0 and 20, which means that x = 10 at the vertex. Using the equation, c(10) = 8140. The coordinates of your parabolic vertex are (10, 8140).
So, the number of thousands of barrels that should be produced to minimize the cost per barrel is 10, and the cost in dollars per barrel when 10 thousand barrels are produced is 8140.
|
|
|
