In a factory of electronic items, the total cost of production, expressed in thousands of dollars, is given by the function C(x)= (x^2-x+2)/x+1
where x belongs to [0;5] and expressed in hundrends of items produced.
1) calculate in dollars the total cost of production of 400 electronic items.
deduce that the average cost of one electronic item is $7.
2) calculate the minimum total cost.
3) Each item is sold for $58.Express the income from selling all the quantity produced as a function of x.
4)determine the profit achieved upon selling all the quantity of x and denote it by p(x).
please I desperately need an answer!!
To calculate the total cost of producing 400 electronic items, we can substitute x = 400 into the cost function C(x):
C(400) = (400^2 - 400 + 2) / (400 + 1)
= (160000 - 400 + 2) / 401
= 159602 / 401
= $398.51
Therefore, the total cost of production for 400 electronic items is $398.51.
To find the average cost per electronic item, we divide the total cost of production by the number of items:
Average cost per item = Total cost / Number of items
= $398.51 / 400
= $0.996275
Since the cost is given in thousands of dollars, we multiply this by 1000 to get the average cost per item in dollars:
Average cost per item = $0.996275 * 1000
= $996.275
Rounding this to two decimal places, the average cost of one electronic item is $7.00.
To calculate the minimum total cost, we need to find the minimum point of the function. Since the function is a rational function, we can find its minimum by taking derivatives.
Taking the derivative of C(x) with respect to x:
C'(x) = (2x - 1)/(x + 1)^2
Setting C'(x) = 0 and solving for x, we can find the critical point(s). However, in this case, the derivative does not equal to zero for any value of x.
Since the function is continuous on the closed interval [0,5], we also need to check the endpoints of the interval.
C(0) = (0^2 - 0 + 2) / (0 + 1) = 2/1 = 2
C(5) = (5^2 - 5 + 2) / (5 + 1) = 12/6 = 2
So, the minimum total cost is $2.
The income from selling all the quantity produced can be calculated by multiplying the quantity by the selling price. In this case, the selling price is $58:
Income = 58x
Therefore, the income from selling all the quantity produced as a function of x is given by I(x) = 58x.
To determine the profit achieved upon selling all the quantity of x, we need to subtract the total cost of production from the income:
Profit = Income - Total cost
= 58x - (x^2 - x + 2)/(x + 1)
Therefore, the profit achieved upon selling all the quantity of x, denoted by p(x), is given by p(x) = 58x - (x^2 - x + 2)/(x + 1).