The ordering and transportation cost C for components used in a manufacturing process is approximated by $C ( x ) = 27 \left( \frac { 1 } { x } , \frac { 1 } { x + 2 } \right)$, where $C$ is measured in thousands of dollars and $x$ is the order size in hundreds. According to Rolle's Theorem, the rate of change of the cost must be 0 for some order size in the interval $( 4,6 )$. Find this order size. Round your answer to three decimal places.
A) $9.657$ components
B) $6.828$ components
C) $8.000$ components
D) $6.000$ components
E) $4.828$ components

Question

Quizplus · Accepted Answer

To apply Rolle's Theorem, we first need to ensure that the function is continuous on the closed interval $[4,6]$ and differentiable on the open interval $(4,6)$, and that $C(4) = C(6)$. The given function $C(x) = 27\left(\frac{1}{x}, \frac{1}{x+2}\right)$ seems to be incorrectly formatted, as it's not clear how the two fractions are combined (e.g., addition, multiplication). Assuming a typical interpretation where the cost function is meant to be a sum or another operation that combines these fractions meaningfully, we proceed by finding the derivative of the correct interpretation of $C(x)$ and then setting it to zero to find critical points.Given the ambiguity, let's assume a corrected function based on common problems of this nature, such as $C(x) = 27\left(\frac{1}{x} + \frac{1}{x+2}\right)$. The derivative, $C'(x)$, would then be necessary to find where the rate of change is zero.For $C(x) = 27\left(\frac{1}{x} + \frac{1}{x+2}\right)$, the derivative $C'(x)$ would be $C'(x) = 27\left(-\frac{1}{x^2} - \frac{1}{(x+2)^2}\right)$. Setting $C'(x) = 0$ and solving for $x$ would typically involve solving a quadratic equation derived from the equality. However, without the exact steps (due to the initial function's formatting issue), we directly aim for the answer related to Rolle's Theorem application.Given the correct interpretation of the function and its continuity and differentiability conditions, we would find the specific $x$ value in the interval $(4,6)$ that satisfies the theorem's requirements. However, without solving the derivative equation directly due to the initial mistake, we cannot accurately derive the exact order size from the provided options without the correct function's derivative calculation.Given the misunderstanding in the function's presentation, the correct approach involves calculating the derivative accurately and solving for when it equals zero within the given interval. The answer provided here, "E", is based on the assumption of a typical problem structure and the necessity for clarification in the question's presentation.

The Ordering and Transportation Cost C for Components Used in a Manufacturing