The total cost, c, in dollars of operating a factory that produces kitchen utensils is C(x)=0.5x^2+40x+8000, where x is the number of items produced in thousands.
a)Determine the marginal cost of producing 5000itmes and compare this with the actual cost of producing the 5001st item.
Ans: C'(5)= 5+40=45 and the comparison is C(5.001)-c(5)=-.045
b) The average cost is found by dividing the total cost by the number of items produced. Determine the average cost of producing 5000items.
Ans: avg cost = total cost/5000
=$1.64
c)Determine the rate of change of the average cost of producing 5000 items. Interpret this value.
Ans: i tried to finding the derivative of the avg cost but i don't get the required ans -0.32
c) (unless I am not reading this correctly)
according to the definition
avg(x) = (0.5x^2+40x+8000)/x
= .5x + 40 + 8000/x
avg '(x) = .5 - 8000/x^2
avg '(5000) = .5 - 8000/5000^2 = .4998
x is the number of items produced in thousands
To find the average cost at 5000 items, we divide the total cost by the number of items (in thousands):
Average Cost = Total Cost / Number of Items
In this case, the number of items is 5000 (in thousands). The total cost can be found by substituting x = 5 into the cost function C(x):
C(5) = 0.5(5^2) + 40(5) + 8000
= 0.5(25) + 200 + 8000
= 12.5 + 200 + 8000
= 8212.5
Therefore, the average cost of producing 5000 items is:
Average Cost = 8212.5 / 5000
= $1.64
Now, to determine the rate of change of the average cost of producing 5000 items, we need to find the derivative of the average cost function. The average cost function is given by:
Average Cost = C(x) / x
Where C(x) is the cost function, and x is the number of items in thousands.
Differentiating the average cost function with respect to x, we get:
d(Average Cost) / dx = (dC(x) / dx - C(x)) / x^2
Substituting x = 5 into this expression, we have:
d(Average Cost) / dx = (C'(5) - C(5)) / (5^2)
To find C'(5), we differentiate the cost function C(x) with respect to x:
C'(x) = dC(x) / dx = 1x^2 + 40x + 0
Therefore,
C'(5) = 1(5^2) + 40(5) + 0
= 25 + 200
= 225
Substituting the values into the expression for the rate of change of the average cost:
d(Average Cost) / dx = (225 - 8212.5) / (5^2)
= (-7987.5) / 25
= -319.5
So, the rate of change of the average cost of producing 5000 items is -319.5. This means that for each additional item produced, the average cost would decrease by $319.5.
To determine the rate of change of the average cost of producing 5000 items, we need to find the derivative of the average cost function.
The average cost, AC(x), is given by dividing the total cost, C(x), by the number of items produced, x:
AC(x) = C(x) / x
The total cost function, C(x), is given as C(x) = 0.5x^2 + 40x + 8000. To find the derivative of the average cost function, we need to differentiate both the numerator and denominator separately.
1. Differentiating the numerator:
To differentiate 0.5x^2 + 40x + 8000 with respect to x, we can apply the power rule and sum rule of differentiation:
dC(x)/dx = d(0.5x^2)/dx + d(40x)/dx + d(8000)/dx
= (0.5 * 2x) + 40 + 0
= x + 40
2. Differentiating the denominator:
The derivative of x is 1.
Now, we can find the derivative of the average cost function:
d(AC(x))/dx = (x + 40) / x
To determine the rate of change of the average cost of producing 5000 items, we substitute x = 5000 into the derivative:
d(AC(x))/dx = (5000 + 40) / 5000
= 5040 / 5000
= 1.008
Interpretation:
The rate of change of the average cost of producing 5000 items is approximately 1.008. This means that for each additional item produced beyond the initial 5000 items, the average cost increases by approximately $1.008.