The daily cost C, in RM, of producing a product is
C(x)=1000+72x-0.06x^2
(0 more & equal x less & equal 60)
where x represents the number of product produced.
a) Find the daily cost of producing 50 units
b) Find the marginal cost function.
c) Find C’(50) and interpret its meaning.
d) Use the marginal cost to estimate the cost of producing 51 units.
e) Find the actual cost of producing 51 units. Compare the actual cost of making 51 units to the estimated cost of producing 51 units found in part d).
f) Find the average cost of producing 51 units.
i need help for question c), d) & e).
Because at c) i don't know what to intrepret. at d) my ans is not same with the given ans. at d) i don't know what to say when comparing.
given ans.
8.a) 4450 b) 72 – 0.12x c) 66 d) 4516 e) 4514.94 f) 88.55
To answer questions c), d), and e), we need to understand the concepts of marginal cost and how it relates to the cost function.
c) To find C'(50) and interpret its meaning:
C'(x) represents the derivative of the cost function C(x). In other words, it gives us the rate of change of the cost with respect to the number of products produced. To find C'(50), we take the derivative of the given cost function C(x) and evaluate it at x = 50.
The cost function C(x) is given as:
C(x) = 1000 + 72x - 0.06x^2
To find C'(x), we differentiate the function:
C'(x) = 72 - 0.12x
Now, evaluate C'(50):
C'(50) = 72 - 0.12(50) = 72 - 6 = 66
Interpretation: C'(50) = 66 means that for each additional unit produced beyond 50, the cost is increasing by RM 66.
d) To use the marginal cost to estimate the cost of producing 51 units:
The marginal cost function gives us the additional cost incurred when producing one more unit. The given marginal cost function is:
Marginal Cost (MC) = 72 - 0.12x
Plug in x = 51 to estimate the cost of producing 51 units:
MC(51) = 72 - 0.12(51) = 72 - 6.12 = 65.88
So, the estimated cost of producing 51 units is RM 65.88.
e) To find the actual cost of producing 51 units and compare it with the estimated cost:
To find the actual cost, substitute x = 51 into the original cost function C(x):
C(51) = 1000 + 72(51) - 0.06(51^2) = 1000 + 3672 - 0.06(2601) ≈ 4514.94
The actual cost of producing 51 units is approximately RM 4514.94.
Comparing the estimated cost (from part d) of RM 65.88) with the actual cost of RM 4514.94, we can see that the estimated cost is lower than the actual cost. This difference can be due to the approximation made when using the marginal cost function. The actual cost may include other factors or additional costs not accounted for in the estimation.
I hope this helps clarify c), d), and e). Let me know if you have any additional questions!