In deciding whether o r not to setup a new manufacturing plant, analysts for a popcorn company have decided that a linear function is a reasonable estimations for the total cost C(x) in dollars to produce x bags of microwave popcorn. They estimate the cost to produce 10000 bags as $5380 and the cost to produce 15000 bags as $7690. Find the marginal cost of the bags of microwave popcorn to be produced in this plant

C(x)=ax+b

a*10000+b=5380
a*15000+b=7690

To find the marginal cost, we need to find the slope (or rate of change) of the linear function.

Step 1: Identify the given information:
The cost to produce 10,000 bags is $5,380 (C1 = 5380).
The cost to produce 15,000 bags is $7,690 (C2 = 7690).

Step 2: Determine the change in cost (ΔC) and the change in quantity (Δx):
ΔC = C2 - C1 = 7690 - 5380 = 2310 (change in cost)
Δx = x2 - x1 = 15000 - 10000 = 5000 (change in quantity)

Step 3: Calculate the marginal cost (M):
M = ΔC / Δx = 2310 / 5000

Step 4: Simplify the result:
M = 0.462

Therefore, the marginal cost of producing bags of microwave popcorn in this plant is $0.462 per bag.

To find the marginal cost, we need to find the derivative of the cost function with respect to the number of bags produced. Since the analysts have determined that a linear function is a reasonable estimate, we can write the cost function as:

C(x) = mx + b,

where x represents the number of bags of microwave popcorn produced, and m and b are constants that need to be determined.

We are given two data points to help us find the values of m and b:

C(10000) = $5380,
C(15000) = $7690.

Substituting these values into the cost function, we can form the following system of equations:

5380 = m(10000) + b,
7690 = m(15000) + b.

To find the values of m and b, we can solve this system of equations. Subtracting the first equation from the second, we have:

7690 - 5380 = m(15000) - m(10000) + b - b,
2310 = 5000m.

Simplifying, we find:

m = 2310 / 5000 = 0.462.

Substituting this value back into the first equation, we can solve for b:

5380 = 0.462(10000) + b,
5380 = 4620 + b,
b = 760.

Therefore, the linear function that represents the total cost to produce x bags of microwave popcorn is:

C(x) = 0.462x + 760.

To find the marginal cost, we need to find the derivative of the cost function. Differentiating the cost function with respect to x, we have:

C'(x) = 0.462.

Therefore, the marginal cost of the bags of microwave popcorn to be produced in this plant is $0.462 per bag.