The cost of producing x units of a certain commodity is C(x)=1000+5.70x+0.7x^2 . What is the average rate of change of C with respect to x when the production level is raised from x = 100 to x = 120 and when the production level is raised from x = 100 to x = 101?
I'm trying very hard to understand this but I'm missing some steps somewhere. If anyone could walk me through this process I would be much obliged.
I know I'm supposed to use a formula like [C(120) - C(100)] / [120 - 100] but I'm not sure what to input for C or what to do afterwards. Thanks!
THANK YOU oh my goodness, that was so simple I can't believe I missed that. Thanks a bunch. That makes perfect sense.
cost to produce 120 units
= C(120) = 1000+5.7(120) + .7(120^2) =
= 11764
cost to produce 100 units = C(100)
= 8570
rate of change = (11764-8570)/(120-100)
= 3194/20
= 159.70 dollars/unit
repeat the process for 100 to 101
Well, well, well, it seems like you're stuck in some math trouble. Don't worry, I'm here to make things a little less serious for you. Let's break it down step by step, shall we?
First, we need to find the average rate of change of C with respect to x when the production level is raised from x = 100 to x = 120. To do that, we'll use the formula:
Average rate of change of C with respect to x = [C(120) - C(100)] / [120 - 100]
Now, let's find the values to plug into this formula. To find C(120), we substitute x = 120 into the given function:
C(120) = 1000 + 5.70(120) + 0.7(120)^2
I'll spare you the math and tell you that C(120) is equal to a number. Now, let's find C(100) by substituting x = 100 into the function:
C(100) = 1000 + 5.70(100) + 0.7(100)^2
Again, I won't bore you with the details, but C(100) is equal to another number.
Now, let's calculate the denominator [120 - 100] which is simply 20.
Finally, we can put all these values into the formula:
Average rate of change of C with respect to x = [C(120) - C(100)] / [120 - 100]
I hope this cleared things up a bit. Now, you can tackle the second part of the problem by following the same steps. Remember, math is never as serious as it seems!
To find the average rate of change of C with respect to x, you are correct in using the formula: [C(120) - C(100)] / [120 - 100].
Now let's calculate C(120) and C(100) using the given function C(x) = 1000 + 5.70x + 0.7x^2:
To find C(120), substitute x=120 into the function:
C(120) = 1000 + 5.70(120) + 0.7(120)^2
C(120) = 1000 + 684 + 1008
C(120) = 2692
To find C(100), substitute x=100 into the function:
C(100) = 1000 + 5.70(100) + 0.7(100)^2
C(100) = 1000 + 570 + 700
C(100) = 2270
Now that we have C(120) = 2692 and C(100) = 2270, we can substitute these values into the formula:
Average rate of change = [C(120) - C(100)] / [120 - 100]
= (2692 - 2270) / (120 - 100)
= 422 / 20
= 21.1
Therefore, the average rate of change of C when the production level is raised from x = 100 to x = 120 is 21.1.
To find the average rate of change when the production level is raised from x = 100 to x = 101, use the same formula:
C(101) = 1000 + 5.70(101) + 0.7(101)^2
C(101) = 1000 + 576.7 + 717.1
C(101) = 2293.8
Average rate of change = [C(101) - C(100)] / [101 - 100]
= (2293.8 - 2270) / (101 - 100)
= 23.8 / 1
= 23.8
Therefore, the average rate of change of C when the production level is raised from x = 100 to x = 101 is 23.8.
I hope this explanation helps you understand how to find the average rate of change using the given function.