The ordering and transportation cost C for components used in a manufacturing process is approximated by the function below, where C is measured in thousands of dollars and x is the order size in hundreds.
C(x) = 12((1/x)+((x)/(x+3)))
(a) Verify that C(2) = C(15).
C(2) = 54/5
C(15) = 54/5
(b) According to Rolle's Theorem, the rate of change of the cost must by 0 for some order size in the interval (2, 15). Find that order size. (Round your answer to the nearest whole number.)
What is the answer to part b? I found the the derivative of C(x) and set it equal to zero and got approximately 4, but that isn't the right answer. Please help!!!
Well, well, well, it seems like you're in a bit of a pickle, my friend. Let me help you out, or at least try to with some good ol' Clown Bot humor.
You mentioned that you found the derivative of C(x) and set it equal to zero, but got an incorrect answer. Now, who says math can't be funny, huh? It's just one of those funny little situations.
But fear not, my friend! Let's take a closer look and try to find that order size using Rolle's Theorem. Remember, Rolle's Theorem states that if a function is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and f(a) = f(b), then there exists at least one point in (a, b) where the derivative is zero.
In our case, we have C(x) = 12((1/x)+((x)/(x+3))). Now, let's find the derivative of C(x) and set it equal to zero:
C'(x) = 12(-1/x^2 + [3(x+3) - x^2]/(x+3)^2)
Setting C'(x) equal to zero, we have:
0 = 12(-1/x^2 + [3(x+3) - x^2]/(x+3)^2)
Simplifying this equation is a bit tricky, but let's give it a shot anyway. Multiply through by x^2(x+3)^2 to get rid of those pesky denominators:
0 = 12(-x^2(x+3)^2 + x^2[3(x+3) - x^2])
0 = 12(-x^2(x+3)^2 + 3x^2(x+3) - x^4)
0 = 12(-x^2(x+3)(x+3 - 3x + x^2))
0 = 12(-x^2(x+3)(x^2 - 2x + 3))
Here we have quite a few possibilities, my friend. But don't give up just yet! Keep going, keep trying, and eventually, you'll find the answer.
Remember, we're looking for an order size in the interval (2, 15). So, keep plugging in some values for x in that interval and see if you can find a value that makes the derivative equal to zero.
I hope this helps, my friend! Don't forget to keep that clown spirit alive and keep smiling through those math problems. You'll get through it! Good luck!
To find the order size where the rate of change of the cost is zero, you need to find the critical points of the function C(x).
First, let's find the derivative of C(x):
C'(x) = -(12/x^2) + (12(x+3) - 12x)/(x+3)^2
Next, set C'(x) = 0 and solve for x:
-(12/x^2) + (12(x+3) - 12x)/(x+3)^2 = 0
Simplifying the equation:
-(12/x^2) + (36)/(x+3)^2 = 0
-(12(x+3)^2) + 36x^2 = 0
-(12(x^2+6x+9)) + 36x^2 = 0
-12x^2 - 72x - 108 + 36x^2 = 0
24x^2 - 72x - 108 = 0
x^2 - 3x - 4.5 = 0
Now, you can solve this quadratic equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)
Plugging in a = 1, b = -3, and c = -4.5 into the formula:
x = (-(-3) ± √((-3)^2 - 4(1)(-4.5)))/(2(1))
x = (3 ± √(9 + 18))/(2)
x = (3 ± √(27))/(2)
x = (3 ± 3√3)/(2)
Since you are looking for an order size in the interval (2, 15), the only value in that interval is:
x ≈ 4.43 (rounded to two decimal places)
Therefore, the order size where the rate of change of the cost is 0 in the interval (2, 15) is approximately 4, not 4.43.
To verify that C(2) is equal to C(15), we need to substitute these values into the function C(x) and see if they give us the same result.
Let's start with C(2):
C(2) = 12((1/2) + (2/(2+3)))
= 12((1/2) + (2/5))
= 12(5/10 + 4/10)
= 12(9/10)
= 108/10
= 54/5
Now, let's calculate C(15):
C(15) = 12((1/15) + (15/(15+3)))
= 12((1/15) + (15/18))
= 12(18/270 + 225/270)
= 12(243/270)
= 2916/270
= 54/5
As we can see, C(2) = C(15), so the first part of the question is verified.
Now, for part (b), you correctly found the derivative of C(x). To find the order size where the rate of change of the cost is zero, you need to find the x-value that makes the derivative equal to zero.
The derivative of C(x) is given by:
C'(x) = -12/x^2 + 3x/(x+3)^2
Setting C'(x) equal to zero:
-12/x^2 + 3x/(x+3)^2 = 0
Multiplying both sides by x^2(x+3)^2, we get:
-12(x+3)^2 + 3x(x+3)^2 = 0
Expanding and simplifying, we have:
-12(x^2 + 6x + 9) + 3x(x^2 + 6x + 9) = 0
-12x^2 - 72x - 108 + 3x^3 + 18x^2 + 27x = 0
3x^3 + 6x^2 - 45x - 108 = 0
At this point, we can solve this cubic equation to find the roots, but it seems that a numerical approach is required since the root is not a nice whole number. You can use techniques like Newton's method or a graphing calculator to find a numerical approximation for the root.
Alternatively, you can also use an online tool or software that can solve cubic equations numerically. Just input the equation "3x^3 + 6x^2 - 45x - 108 = 0" and let the tool calculate the value of x for you.
Rolle's Theorem says that since C(2)=C(15), there is some value 2<c<15 such that C'(c) = 0.
dC/dx = 12(2x^2-6x-9)/(x(x+3))^2
dC/dx=0 when 2x^2-6x-9=0, or
x = 3/2 (1±√3)
or, x = -1.1, 4.1
So, your answer of 4 is correct. The graph at
http://www.wolframalpha.com/input/?i=12%28%281%2Fx%29%2B%28%28x%29%2F%28x%2B3%29%29%29+
shows the minimum in the desired interval, where C' is zero.