The marginal cost of a product is modeled by:
14/cube root of 14x+5. Where x is the number of units. X=15, C=120.
a) Find the cost function C=
b) Find the Cost in dollars of producing 80 units
I am lost as how to get the fraction at the beginnning of the next step, What is the trick, or rule that I can't seem to learn>!>!!!?????
To find the cost function (C), you need to integrate the given marginal cost (MC) function. In this case, the marginal cost function is:
MC = 14/(∛(14x + 5))
To find the cost function, you need to integrate the marginal cost function with respect to x. Here's how you can do it step by step:
Step 1: Rewrite the marginal cost function in a slightly different form:
MC = 14(14x + 5)^(-1/3)
Step 2: To integrate the function, you can use the power rule of integration. The power rule states that the integral of x^n with respect to x is given by [(x^(n+1))/(n+1)].
Step 3: Apply the power rule to integrate the marginal cost function:
C(x) = integral of MC dx
= integral of 14(14x + 5)^(-1/3) dx
= 14 * [(14x + 5)^(-1/3 + 1)] / (-1/3 + 1) + C1
= 14 * [(14x + 5)^(2/3)] / (2/3) + C1
= (21/2) * (14x + 5)^(2/3) + C1
Here, C1 is the constant of integration. To determine C1, use the given information that when x = 15, C = 120. Plug these values into the cost function and solve for C1:
120 = (21/2) * (14(15) + 5)^(2/3) + C1
120 = (21/2) * 5041^(2/3) + C1
Solving for C1:
C1 = 120 - (21/2) * 5041^(2/3)
Now, we have the cost function C(x) in terms of x.
To find the cost in dollars of producing 80 units, substitute x = 80 into the cost function C(x):
C(80) = (21/2) * (14(80) + 5)^(2/3) + C1
Now, plug in the value of C1 we found earlier to get the final answer.
Dc/dx=14/³√(14x+5)
C=14∫1/³√(14x+5)dx
Put u=14x+5
Du/dx=14
Dx=du/14
C=14∫(u^-1/3)du/14=14/14(u^(2/3)/(2/3))+c
C=3/2(³√(14x+5))²+c
120=3/2(³√(14(15)+5))²+c
2/3(120)=(³√((5.99))²+c
80=35.8+c
c=80-35.8=34.2
C=3/2(³√(14(80)+5))²+34.2=3/2(108.16)+34.2=162.24+34.2=196.44
Check the maths for possible typo