In a certain factory, output Q is related to inputs x and y by the equation $Q = 3 x ^ { 3 } + 5 x ^ { 2 } y ^ { 2 } + 2 y ^ { 3 }$ . If the current levels of input are x = 255 and y = 155, use calculus to estimate the change in input y that should be made to offset a decrease of 0.6 unit in input x so that output will be maintained at its current level. Round your answer to two decimal places, if necessary.
A) An increase of 0.37
B) A decrease of 0.37
C) It cannot be determined
D) No change

Question

Quizplus · Accepted Answer

To maintain the output level, we use the concept of total differential: $dQ = \frac{\partial Q}{\partial x} dx + \frac{\partial Q}{\partial y} dy = 0$. Calculate $\frac{\partial Q}{\partial x}$ and $\frac{\partial Q}{\partial y}$, substitute $x = 255$, $y = 155$, and $dx = -0.6$ to solve for $dy$. The result is approximately 0.37, indicating an increase in y.

In a Certain Factory, Output Q Is Related to Inputs Q=3x3+5x2y2+2y3Q = 3 x ^ { 3 } + 5 x ^ { 2 } y ^ { 2 } + 2 y ^ { 3 }Q=3x3+5x2y2+2y3

In a Certain Factory, Output Q Is Related to Inputs $Q = 3 x ^ { 3 } + 5 x ^ { 2 } y ^ { 2 } + 2 y ^ { 3 }$