on capital invested by a manufacturer can be modeled by the equation \( 60 x^{\frac{3}{4}} y^{\frac{3}{4}}=3240 \). a) Find \( \frac{d y}{d x} \) and evaluate at the point \( (81,16) \). b) Interpret the result of part a.

To find \( \frac{dy}{dx} \) for the equation \( 60 x^{\frac{3}{4}} y^{\frac{3}{4}} = 3240 \), first rearrange the equation into a suitable form. Divide both sides by 60: \[ x^{\frac{3}{4}} y^{\frac{3}{4}} = 54 \] Now, we can differentiate both sides implicitly with respect to \( x \): \[ \frac{3}{4} x^{-\frac{1}{4}} y^{\frac{3}{4}} + x^{\frac{3}{4}} \cdot \frac{3}{4} y^{-\frac{1}{4}} \frac{dy}{dx} = 0 \] Next, isolate \( \frac{dy}{dx} \): \[ \frac{3}{4} y^{-\frac{1}{4}} \frac{dy}{dx} = -\frac{3}{4} x^{-\frac{1}{4}} y^{\frac{3}{4}} \] \[ \frac{dy}{dx} = -\frac{y^{\frac{1}{4}}}{x^{\frac{1}{4}}} \] Now, we can substitute \( x = 81 \) and \( y = 16 \): \[ \frac{dy}{dx} = -\frac{16^{\frac{1}{4}}}{81^{\frac{1}{4}}} = -\frac{2}{3} \] So, at the point \( (81, 16) \), \( \frac{dy}{dx} = -\frac{2}{3} \). Interpreting this result, \( \frac{dy}{dx} = -\frac{2}{3} \) indicates that for every increase of 3 units in \( x \), \( y \) decreases by 2 units. This negative slope suggests that there is an inverse relationship between capital \( y \) and production amount \( x \) at this specific point, hinting at diminishing returns on capital investment for this manufacturer. In simple terms, as they invest more in manufacturing, there’s a slight decrease in the investment needed elsewhere, reflecting efficiency in their operational spending.