Question

The marginal cost function, in dollars per item, for producing the x th item of a certain brand of bar stool is given by MC(x)=20−0. 5 x , 0≤ x≤ 100. The fixed cost is $200. Estimating the total cost of producing 100 bars tools using the left-rectangle approximation with five rectangles, we conclude that the total cost is approximately $

Answer 1

Using the left-rectangle approximation with five rectangles, the total cost of producing 100 bar stools is estimated to be $200.

Step-by-step explanation:

Using the left-rectangle approximation with five rectangles, we can estimate the total cost of producing 100 bar stools using the given marginal cost function. The left-rectangle approximation involves dividing the interval [0,100] into five equal subintervals and approximating the area under the curve of the marginal cost function within each subinterval as the height of the rectangle multiplied by the width of the subinterval.

In this case, the width of each subinterval would be 100/5 = 20. The height of each rectangle would be the value of the marginal cost function at the left endpoint of the subinterval. So, the five rectangles would have heights MC(0), MC(20), MC(40), MC(60), and MC(80) respectively. To estimate the total cost, we sum up the areas of these rectangles. The formula for the area of a rectangle is height multiplied by width.

Given that the marginal cost function is MC(x) = 20 - 0.5x and 0 ≤ x ≤ 100, we can substitute the values of x into the marginal cost function to find the heights of the rectangles:

1. Height of the first rectangle = MC(0) = 20 - 0.5(0) = 20
2. Height of the second rectangle = MC(20) = 20 - 0.5(20) = 10
3. Height of the third rectangle = MC(40) = 20 - 0.5(40) = 0
4. Height of the fourth rectangle = MC(60) = 20 - 0.5(60) = -10
5. Height of the fifth rectangle = MC(80) = 20 - 0.5(80) = -20

Since the width of each rectangle is 20, we can calculate the areas:

1. Area of the first rectangle = 20 × 20 = 400
2. Area of the second rectangle = 10 × 20 = 200
3. Area of the third rectangle = 0 × 20 = 0
4. Area of the fourth rectangle = -10 × 20 = -200
5. Area of the fifth rectangle = -20 × 20 = -400

To estimate the total cost, we sum up the areas of the rectangles:

Total cost = (400 + 200 + 0 - 200 - 400) + fixed cost = 0 + 200 = $200

Answer 2

The total cost for producing 100 bar stools, using the marginal cost function MC(x) = 20 - 0.5x and a fixed cost of $200, is estimated to be approximately $800 using the left-rectangle approximation method for the intervals where marginal costs are positive.

Step-by-step explanation:

The question pertains to the calculation of the total cost of producing a good using marginal costs and includes fixed costs. Since the marginal cost function provided is MC(x) = 20 - 0.5x, and the fixed cost is $200, we can estimate the total cost of production using the left-rectangle approximation method with five rectangles. The width of each rectangle would be 100/5 = 20, as we are producing 100 items and we divide this into five rectangles. So, for each of the five intervals (0-20, 20-40, 40-60, 60-80, 80-100), we calculate the area of each rectangle using the marginal cost at the start of each interval and sum them all up, adding the fixed cost afterwards to estimate the total cost.

To illustrate:

First interval (0-20): MC(0) = $20

Second interval (20-40): MC(20) = $20 - (0.5 × 20) = $10

Third interval (40-60): MC(40) = $20 - (0.5 × 40) = $0 (as the production might not be economically viable beyond this point)

Fourth interval (60-80): MC(60) = $20 - (0.5 × 60) = -$10 (costs become negative, not realistic in this context)

Fifth interval (80-100): MC(80) = $20 - (0.5 × 80) = -$20 (again, not realistic)

However, since the marginal cost cannot be negative in this context, we should reconsider the values and possibly only use the intervals where the marginal cost is positive.

Assuming we only consider positive marginal costs, the total cost (TC) can be estimated as the sum of the areas of the rectangles that represent the marginal cost of each interval multiplied by the quantity (20 units per interval), plus the fixed costs.

TC = Fixed Cost + ∑(MC×Quantity) = $200 + (20×$20) + (20×$10) + (20×$0) = $200 + $400 + $200 = $800

Thus, the total cost for producing 100 bar stools with the given marginal cost function is approximately $800.