(1):Given the cost function c(x)=10+2x
and the revenue functio r(x)=3x,find break even quantity and the price
(2):if the marginal cost of producing x unit is given by c'(x)=0.4x^3+2x and the fixed cost is p4000.00 find the cost function c(x) and cost produnction 40unit???
(1): To find the break-even quantity and price, we need to set the cost function equal to the revenue function and solve for x.
The cost function is given by c(x) = 10 + 2x, and the revenue function is given by r(x) = 3x.
Setting the two functions equal, we have:
10 + 2x = 3x
Subtracting 2x from both sides, we get:
10 = x
Therefore, the break-even quantity is 10 units.
To find the price, we can substitute this value back into the revenue function:
r(10) = 3(10)
= 30
Therefore, the price at the break-even quantity is $30.
(2): To find the cost function c(x) and the cost of producing 40 units, we need to integrate the marginal cost function.
The marginal cost function is given by c'(x) = 0.4x^3 + 2x.
To find the cost function, we integrate c'(x) with respect to x:
∫ c'(x) dx = ∫ (0.4x^3 + 2x) dx
Integrating term by term, we get:
c(x) = 0.1x^4 + x^2 + C
Where C is the constant of integration.
Given that the fixed cost is p4000.00, we can substitute this into the cost function to find the value of C:
c(0) = 4000
Substituting x = 0 and c(x) = 4000, we get:
0.1(0)^4 + (0)^2 + C = 4000
Simplifying, we have:
C = 4000
Therefore, the cost function c(x) is:
c(x) = 0.1x^4 + x^2 + 4000
To find the cost of producing 40 units, we substitute x = 40 into the cost function:
c(40) = 0.1(40)^4 + (40)^2 + 4000
Simplifying, we get:
c(40) = 32000 + 1600 + 4000
= 37600
Therefore, the cost of producing 40 units is p37600.00.
(1): To find the break-even quantity and the price, we need to set the cost function equal to the revenue function and solve for x.
First, let's set c(x) equal to r(x):
10 + 2x = 3x
Now, we can solve for x:
10 = 3x - 2x
10 = x
So, the break-even quantity is x = 10.
To find the price, we can substitute x = 10 into the revenue function:
r(x) = 3x
r(10) = 3(10)
r(10) = 30
Therefore, the price is 30.
(2): To find the cost function c(x) and the cost of producing 40 units, we need to integrate the marginal cost function.
First, let's find the cost function c(x) by integrating c'(x):
c(x) = ∫(c'(x)) dx
c(x) = ∫(0.4x^3 + 2x) dx
Integrating term by term, we get:
c(x) = 0.1x^4 + x^2 + C
Since we know the fixed cost is p4000.00, we can determine the constant C:
c(0) = 4000
0.1(0)^4 + (0)^2 + C = 4000
C = 4000
So, the cost function c(x) is:
c(x) = 0.1x^4 + x^2 + 4000
Now, let's find the cost of producing 40 units by plugging x = 40 into c(x):
c(40) = 0.1(40)^4 + (40)^2 + 4000
Calculating this expression, we can find the cost of producing 40 units.