the cost (in dollars) of producing x units of a certain commodity is C(x)=5090+13x+0.09x^2. Find the average rate of change with respect to x when the production level is changed from x=105 to x=107.
eeg
To find the average rate of change with respect to x, we need to calculate the difference in cost (C(x)) and the difference in x (production level) and then divide them.
First, we need to find the cost when x = 105:
C(105) = 5090 + 13(105) + 0.09(105)^2
Next, we need to find the cost when x = 107:
C(107) = 5090 + 13(107) + 0.09(107)^2
Now, we can calculate the average rate of change:
Average rate of change = (C(107) - C(105))/(107 - 105)
Step 1:
C(105) = 5090 + 13(105) + 0.09(105)^2
C(105) = 5090 + 1365 + 992.25
C(105) = 7447.25
Step 2:
C(107) = 5090 + 13(107) + 0.09(107)^2
C(107) = 5090 + 1391 + 1008.99
C(107) = 7489.99
Step 3:
Average rate of change = (C(107) - C(105))/(107 - 105)
Average rate of change = (7489.99 - 7447.25)/(107 - 105)
Average rate of change = 42.74/2
Average rate of change = 21.37
Therefore, the average rate of change with respect to x when the production level is changed from x = 105 to x = 107 is 21.37 dollars.
To find the average rate of change with respect to x, we need to find the change in the cost divided by the change in x. In this case, the change in x is 107 - 105 = 2 units.
First, let's find the cost when the production level is x = 105:
C(105) = 5090 + 13(105) + 0.09(105)^2
C(105) = 5090 + 1365 + 992.25
C(105) = 7447.25
Next, let's find the cost when the production level is x = 107:
C(107) = 5090 + 13(107) + 0.09(107)^2
C(107) = 5090 + 1391 + 1078.83
C(107) = 7560.83
Now we can calculate the average rate of change:
Average rate of change = (C(107) - C(105)) / (107 - 105)
= (7560.83 - 7447.25) / 2
= 113.58 / 2
= 56.79
Therefore, the average rate of change with respect to x when the production level is changed from x = 105 to x = 107 is 56.79 dollars per unit.