|
Question 90413This question is from textbook Beginning Algebra
: If someone could help me with this word problem I would very appreciative.
Business and finance. In planning for a new item, a manufacturer assumes that the number of items produced x and the cost in dollars C of producing these items are related by a linear equation. Projections are that 100 items will cost $10,000 to produce and that 300 items will cost $22,000 to produce. Find the equation that relates C and x.
This question is from textbook Beginning Algebra
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! Let's suppose that we label the y-axis in dollars and the x-axis in items produced.
.
Then our coordinate points will be in the form (x, y) corresponding to (items, dollars).
.
In this form, the problem gives you two points: (100, $10,000) and (300, $22,000).
.
Remember that each of these points is in the form (x,y).
.
Now suppose that we use the slope-intercept form of an equation. The slope intercept
form is:
.
.
in which m is the slope of the graph and b is the value where the graph crosses the y-axis.
This is a linear equation, so it will lead us to a linear relationship between cost (C)
and the number of units produced (x).
.
Let's try to find m and b. Then we can substitute those values into the slope-intercept
form and we will have the equation.
.
The slope is defined as the amount of change in the y direction between the two points divided
by the change in the x direction between the two points. Notice that the change in going
from the y value of the first point to the y value of the second point is an increase
from $10,000 to $22,000 ... an increase of $12,000.
.
Similarly, the corresponding change in going from the x value of the first point to the
x value of the second point is an increase from 100 items to 300 items ... an increase
of 200 items.
.
Therefore, we can say that the slope is the change in y (a change of $12,000) divided by
the change in x (a change of 200 items). Therefore, the slope is:
.
.
Now that we know the slope (m) is 60, we can go back to the slope-intercept form of the
equation and substitute 60 for m to get:
.
.
At this point, let's recognize that the y-value is the cost of production so we can substitute
C for y. This makes the equation:
.
.
furthermore x represents the number of items produced.
.
All we now need is to find b. We can do that by taking either of the two points and
substituting the x and C values for that point into the equation we now have. Let's choose
the point (100, $10,000). [We could just as well use (300, $22,000)]. Substituting
100 for x and 10000 for C, our equation becomes:
.
.
Multiply the 60 times 100 and you get:
.
.
Finally solve for b by subtracting 6000 from both sides of the equation to get:
.
.
This simplifies to  . Substituting this value of b into our cost equation
results in:
.
.
This should be the equation that you are looking for. Let's check it by taking one of
the coordinates of the second point, plugging it into the equation, and seeing if it
gives the other coordinate value of that point. Recall that we used the point (100, 10000)
to find the value of b. So we can use the second point (300, 22000) in our check.
Let's take the x value of 300 and plug that value into our equation. When we do that the
Cost equation becomes:
.
.
Multiplying on the right side [ 60 times 300] results in:
.
.
Finally adding the terms on the right side results in:
.
.
That's exactly the answer we should get ... for 300 units the cost is $22,000. So the
equation checks out. The equation you are looking for is:
.
.
Hope this helps you to understand the problem.
.
|
|
|
| |
