Let x represent the units of labor and y the capital invested in a manufacturing process. When 135,540 units are produced, the relationship between labor and capital can be modeled by $100 x ^ { 0.75 } y ^ { 0.25 } = 135,540$ . Find the rate of change of y with respect to x when $x = 1500 \text { and } y = 135,540$ .
A)-2
B)0
C)3
D)-7
E)5

Question

Quizplus · Accepted Answer

To find the rate of change of y with respect to x, we differentiate the given equation implicitly with respect to x. The equation is $100 x^{0.75} y^{0.25} = 135,540$. Differentiating both sides with respect to x, we get:  $$ 100 \left(0.75 x^{-0.25} y^{0.25} + x^{0.75} \cdot 0.25 y^{-0.75} \frac{dy}{dx} \right) = 0 $$  Solving for $\frac{dy}{dx}$ when $x = 1500$ and $y = 135,540$, we find $\frac{dy}{dx} = -2$.

Let X Represent the Units of Labor and Y the Capital