Find the cost function if the marginal cost function is given by C'(x)= x^(1/2) +7 and 4 units cost $40.
Well, aren't you in for a treat! Let's have a laugh while we figure this out.
So, the marginal cost function is given by C'(x) = x^(1/2) +7. We know that the cost of 4 units is $40.
Now, to find the cost function, we need to integrate the marginal cost function. So, let's put on our math hats (or should I say math wigs?).
Integrating C'(x), we get C(x) = ∫(x^(1/2) + 7) dx. And remember, integration is just fancy math for "finding the antiderivative".
So, integrating x^(1/2), we get (2/3)x^(3/2). And integrating 7, we get 7x.
Putting it all together, we have C(x) = (2/3)x^(3/2) + 7x + C, where C is the constant of integration.
But wait, we have one more piece of the puzzle. We know that the cost of 4 units is $40. So, we can plug in the values and solve for C.
Using C(4) = 40, we get: (2/3)(4)^(3/2) + 7(4) + C = 40. Simplifying, we have (8/3)√4 + 28 + C = 40.
Now, let's solve for C. I'll leave the math to you, my friend. Just remember to keep a smile on your face while crunching those numbers!
And voila! You have your cost function. Just remember that my humor is priceless, even if the cost function does have a price tag!
To find the cost function given the marginal cost function, you need to integrate the marginal cost function with respect to x. Let's go through the steps to find the cost function.
Step 1: Integrate the marginal cost function with respect to x:
∫ C'(x) dx = ∫ (x^(1/2) + 7) dx
Step 2: Integrate each term separately:
∫ x^(1/2) dx = (2/3) * x^(3/2)
∫ 7 dx = 7x
Step 3: Combine the results:
Cost function C(x) = (2/3) * x^(3/2) + 7x + C
Step 4: Use the given information to find the constant C. We know that when 4 units are produced, the cost is $40. So, we can substitute this value into the cost function:
C(4) = (2/3) * 4^(3/2) + 7 * 4 + C = 40
Simplifying this equation:
(2/3) * 8 + 28 + C = 40
(16/3) + 28 + C = 40
(16/3) + 84/3 + C = 40
100/3 + C = 40
C = 40 - 100/3
C = (120 - 100) / 3
C = 20/3
Step 5: Substitute the value of C back into the cost function:
Cost function C(x) = (2/3) * x^(3/2) + 7x + 20/3
Therefore, the cost function is C(x) = (2/3) * x^(3/2) + 7x + 20/3.
To find the cost function, we need to integrate the marginal cost function.
First, let's integrate the marginal cost function C'(x) = x^(1/2) + 7 with respect to x:
∫ C'(x) dx = ∫ (x^(1/2) + 7) dx
Using the power rule of integration, we can integrate x^(1/2) and 7 separately:
∫ x^(1/2) dx = (2/3) x^(3/2) + C1
∫ 7 dx = 7x + C2
Putting it all together, we get:
C(x) = (2/3) x^(3/2) + 7x + C
Now, we need to solve for C by using the given information that 4 units cost $40. We can substitute x = 4 and C(x) = 40 into the cost function:
40 = (2/3)(4)^(3/2) + 7(4) + C
Simplifying this equation:
40 = (2/3)(8) + 28 + C
40 = 16/3 + 28 + C
40 = 64/3 + C
Now, let's solve for C:
C = 40 - 64/3
C = (120 - 64)/3
C = 56/3
Therefore, the cost function is:
C(x) = (2/3) x^(3/2) + 7x + 56/3
C'(x)= x^(1/2) +7
integrate:
C(x) = (2/3)x^(3/2) + 7x + k, where k is a constant,
given: when x = 4, C(4) = 40
40 = (2/3)(4^(3/2) ) + 28 + K
40 = 16/3 + 28 + k
finish it up.