In your new job as a strategic planner at BusCalcInc, you have been asked to evaluate a proposal for manufacturing a new product. Other analysts have determined that a linear function is a reasonable estimate for the total cost C(x) in dollars to produce x items. Another team has determined that the cost to produce 1,000 items is $2,110,000 and the cost to produce 5,000 items is $12,550,000. A) Write the linear function C(x) that describes the cost function? B) Find the total cost to produce 50,000 items? C) Find the marginal cost of the new product (instantaneous rate of change.)
your linear function will look like
C(x) = mx + b
you are given 2 ordered pairs
(1000,2110000) and (5000,12550000)
Since you are in Calculus you have to know how to find the eqution of a straight line if you are given two points.
Hint: first find the slope, that gives you the m value
etc.
A) To find the linear function C(x) that describes the cost function, we can use the given data points of producing 1,000 items and 5,000 items.
Let's assume the linear function is of the form C(x) = mx + b, where m is the slope and b is the y-intercept.
Using the given data points:
C(1,000) = $2,110,000
C(5,000) = $12,550,000
Substituting these values into the linear function, we get two equations:
2,110,000 = m(1,000) + b
12,550,000 = m(5,000) + b
We can solve these equations simultaneously to find the values of m and b.
Rearranging the first equation, we have:
2,110,000 = 1,000m + b -----(i)
Rearranging the second equation, we have:
12,550,000 = 5,000m + b -----(ii)
Now, subtracting equation (i) from equation (ii), we get:
12,550,000 - 2,110,000 = 5,000m - 1,000m + b - b
10,440,000 = 4,000m
Dividing both sides by 4,000, we find:
m = 10,440,000 / 4,000
m = 2,610
Now, substituting the value of m into equation (i), we can solve for b:
2,110,000 = 1,000(2,610) + b
Expanding and simplifying:
2,110,000 = 2,610,000 + b
b = 2,110,000 - 2,610,000
b = -500,000
Therefore, the linear function C(x) is:
C(x) = 2,610x - 500,000
B) To find the total cost to produce 50,000 items, we can substitute x = 50,000 into the linear function C(x) we found in part A:
C(50,000) = 2,610(50,000) - 500,000
Calculating:
C(50,000) = 130,500,000 - 500,000
C(50,000) = $130,000,000
Therefore, the total cost to produce 50,000 items is $130,000,000.
C) The marginal cost of the new product refers to the instantaneous rate of change of the cost function C(x). In other words, it represents how much the cost changes when we produce an additional item.
To find the marginal cost, we need to find the derivative of the cost function C(x). Since the cost function is linear, the derivative will be a constant (the slope of the line).
The derivative of the cost function C(x) = 2,610x - 500,000 with respect to x is simply the coefficient of x, which is 2,610. This represents the marginal cost of the new product.
Therefore, the marginal cost of the new product is $2,610 per item.