If y = c ( x ) represents the cost of producing x units in a manufacturing process, the elasticity of cost is defined as $E ( x ) = \frac { \text { marginal cost } } { \text { average cost } } = \frac { C ^ { \prime } ( x ) } { C ( x ) / x } = \frac { x } { y } \frac { d y } { d x }$. Find the cost function if the elasticity function is $E ( x ) = \frac { 12 x - y } { 2 y - 6 x }$, where $C ( 90 ) = 272$ and $x \geq 90$.
A) $y = \frac { 3 \left( x ^ { 2 } + \sqrt { x ^ { 4 } - 730,210 x } \right) } { x }$

B) $y = \frac { 6 \left( x ^ { 2 } + \sqrt { x ^ { 4 } - 730,210 x } \right) } { x }$

C) $y = \frac { 6 \left( x ^ { 2 } + \sqrt { x ^ { 4 } - 728,960 } \right) } { x }$

D) $y = \frac { 3 \left( x ^ { 2 } + 2 \sqrt { x ^ { 4 } - 734,041 } \right) } { x }$

E)$y = \frac { 3 \left( x ^ { 2 } + \sqrt { x ^ { 4 } - 728,960 x } \right) } { x }$

Question

If  y = c ( x ) represents the cost of producing  x  units in a manufacturing process, the elasticity of cost is defined as $E ( x ) = \frac { \text { marginal cost } } { \text { average cost } } = \frac { C ^ { \prime } ( x ) } { C ( x ) / x } = \frac { x } { y } \frac { d y } { d x }$. Find the cost function if the elasticity function is $E ( x ) = \frac { 12 x - y } { 2 y - 6 x }$, where $C ( 90 ) = 272$ and $x \geq 90$.
A) $y = \frac { 3 \left( x ^ { 2 } + \sqrt { x ^ { 4 } - 730,210 x } \right) } { x }$

B) $y = \frac { 6 \left( x ^ { 2 } + \sqrt { x ^ { 4 } - 730,210 x } \right) } { x }$

C) $y = \frac { 6 \left( x ^ { 2 } + \sqrt { x ^ { 4 } - 728,960 } \right) } { x }$

D) $y = \frac { 3 \left( x ^ { 2 } + 2 \sqrt { x ^ { 4 } - 734,041 } \right) } { x }$

E)$y = \frac { 3 \left( x ^ { 2 } + \sqrt { x ^ { 4 } - 728,960 x } \right) } { x }$

Quizplus · Accepted Answer

Given the elasticity function $E(x) = \frac{12x - y}{2y - 6x}$, we can solve for the cost function $y = C(x)$. The elasticity of cost is defined as $E(x) = \frac{x}{y} \frac{dy}{dx}$, which means we need to equate this with the given elasticity function and solve the resulting differential equation. The correct answer must satisfy the given condition $C(90) = 272$ and the form of the differential equation derived from the given elasticity function. Among the options, only option E correctly represents a solution that can be derived from the given elasticity function and satisfies the initial condition when evaluated.

If Y = C ( X ) Represents the Cost