Question

Suppose total cost in dollars from the production of x printers is given by c(x) = 0.0001x3 + 0.005x2 + 28x + 3000. (a) find the average rate of change of total cost when production changes from 200 to 300 printers. incorrect: your answer is incorrect. (b) find the average rate of change of total cost when production changes from 300 to 500 printers. incorrect: your answer is incorrect.

Answer 1

To find the average rate of change for total costs over different intervals of production, we determine the changes in cost and quantity for the intervals, then divide the change in cost by the change in quantity for both intervals specified.

Step-by-step explanation:

The question involves finding the average rate of change of the total cost function, which is related to the concept of slope in algebra or the derivative in calculus. To find the average rate of change of the total cost when production changes from one quantity to another, we simply calculate the change in cost divided by the change in quantity.

Part (a) Calculation

For the production change from 200 to 300 printers, we calculate the average rate of change as follows:

c(300) = 0.0001(300)^3 + 0.005(300)^2 + 28(300) + 3000,
c(200) = 0.0001(200)^3 + 0.005(200)^2 + 28(200) + 3000,

Subtract c(200) from c(300) and divide by the change in x (300 - 200).

Part (b) Calculation

For the production change from 300 to 500 printers, again:

c(500) = 0.0001(500)^3 + 0.005(500)^2 + 28(500) + 3000,
c(300) = 0.0001(300)^3 + 0.005(300)^2 + 28(300) + 3000,

Subtract c(300) from c(500) and divide by the change in x (500 - 300).

These calculations will give us the average total cost changes for the specified intervals of production, and thus allow us to understand how costs change as production scales.

Answer 2

Determine the rate of change of the total cost by deriving the equation.
dc(x) / dt = 3(0.0001)x² + (2)(0.005)(x) + 28

(a) Substitute 200 and 300 to the equation,
(200) : (3)(0.0001)(200²) + (2)(0.005)(200) + 28 = 42
(300) : (3)(0.0001)(300²) + (2)(0.005)(300) + 28 = 58

The difference between the values is 16.

b. Similarly, substitute 500 to the equation,
(500) : (3)(0.0001)(500²) + (2)(0.005)(500) + 28 = 108
From a, we have 58 for 300. The difference between the values is 50.