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 solve the problem, we will follow the steps outlined: ### Step 1: Extract Known Conditions We have the equation: \[ 60 x^{\frac{3}{4}} y^{\frac{3}{4}} = 3240 \] We need to find \( \frac{dy}{dx} \) and evaluate it at the point \( (81, 16) \). ### Step 2: Differentiate Implicitly We will differentiate both sides of the equation with respect to \( x \). 1. Start with the original equation: \[ 60 x^{\frac{3}{4}} y^{\frac{3}{4}} = 3240 \] 2. Differentiate both sides with respect to \( x \): - Use the product rule on the left side: \[ \frac{d}{dx}(60 x^{\frac{3}{4}} y^{\frac{3}{4}}) = 60 \left( \frac{3}{4} x^{-\frac{1}{4}} y^{\frac{3}{4}} + x^{\frac{3}{4}} \frac{3}{4} y^{-\frac{1}{4}} \frac{dy}{dx} \right) \] - The right side is a constant, so its derivative is 0: \[ 0 \] 3. Set the differentiated equation equal to zero: \[ 60 \left( \frac{3}{4} x^{-\frac{1}{4}} y^{\frac{3}{4}} + x^{\frac{3}{4}} \frac{3}{4} y^{-\frac{1}{4}} \frac{dy}{dx} \right) = 0 \] 4. Simplify: \[ \frac{3}{4} x^{-\frac{1}{4}} y^{\frac{3}{4}} + x^{\frac{3}{4}} \frac{3}{4} y^{-\frac{1}{4}} \frac{dy}{dx} = 0 \] 5. Solve for \( \frac{dy}{dx} \): \[ x^{\frac{3}{4}} \frac{3}{4} y^{-\frac{1}{4}} \frac{dy}{dx} = -\frac{3}{4} x^{-\frac{1}{4}} y^{\frac{3}{4}} \] \[ \frac{dy}{dx} = -\frac{y^{\frac{3}{4}}}{x^{\frac{3}{4}}} \cdot \frac{1}{y} = -\frac{y^{\frac{3}{4} - 1}}{x^{\frac{3}{4}}} \] ### Step 3: Evaluate at the Point \( (81, 16) \) Now we will substitute \( x = 81 \) and \( y = 16 \) into the derivative: \[ \frac{dy}{dx} = -\frac{16^{\frac{3}{4} - 1}}{81^{\frac{3}{4}}} \] Calculating \( 16^{\frac{3}{4} - 1} \) and \( 81^{\frac{3}{4}} \): - \( 16^{\frac{3}{4}} = (2^4)^{\frac{3}{4}} = 2^3 = 8 \) - \( 81^{\frac{3}{4}} = (3^4)^{\frac{3}{4}} = 3^3 = 27 \) Thus: \[ \frac{dy}{dx} = -\frac{8}{27} \] ### Step 4: Interpret the Result The result \( \frac{dy}{dx} = -\frac{8}{27} \) indicates that for every increase of 27 units in \( x \), \( y \) decreases by 8 units. This negative slope suggests an inverse relationship between the capital invested in \( x \) and \( y \) at the point \( (81, 16) \). ### Final Answer a) \( \frac{dy}{dx} = -\frac{8}{27} \) b) The interpretation is that an increase in \( x \) (capital invested) leads to a decrease in \( y \) (another variable related to the investment) at the specified point.