The number of cars produced when \( x \) dollars is spent on labor and \( y \) dollars is spent on capital invested by a manufacturer can be modeled by the equation \( 40 x^{1 / 3} y^{2 / 3}=480 \). (Both \( x \) and \( y \) are measured in thousands of dollars.) a. Find \( \frac{d y}{d x} \) and evaluate at the point \( (64,27) \). b. Interpret the result of a.

**a. Finding \(\frac{dy}{dx}\) and evaluating at \((64,27)\):** We start with the equation \[ 40x^{1/3}y^{2/3} = 480. \] Differentiate both sides with respect to \(x\). First, write the equation implicitly as \[ 40x^{1/3}y^{2/3} - 480 = 0. \] Differentiate term by term. The derivative of \(40x^{1/3}y^{2/3}\) using the product rule is: \[ \frac{d}{dx}\bigl(40x^{1/3}y^{2/3}\bigr) = 40\left(\frac{d}{dx}\bigl(x^{1/3}\bigr)y^{2/3} + x^{1/3}\frac{d}{dx}\bigl(y^{2/3}\bigr)\right). \] Differentiate the factors: - \(\frac{d}{dx}\bigl(x^{1/3}\bigr) = \frac{1}{3}x^{-2/3}\). - \(\frac{d}{dx}\bigl(y^{2/3}\bigr)\) requires the chain rule: \(\frac{d}{dx}\bigl(y^{2/3}\bigr) = \frac{2}{3}y^{-1/3}\frac{dy}{dx}\). Thus, \[ 40\left[\frac{1}{3}x^{-2/3}y^{2/3} + x^{1/3}\frac{2}{3}y^{-1/3}\frac{dy}{dx}\right] = 0. \] Simplify by dividing both sides by 40 and multiplying by 3: \[ x^{-2/3}y^{2/3} + 2x^{1/3}y^{-1/3}\frac{dy}{dx} = 0. \] Solve for \(\frac{dy}{dx}\): \[ 2x^{1/3}y^{-1/3}\frac{dy}{dx} = -x^{-2/3}y^{2/3}, \] \[ \frac{dy}{dx} = -\frac{x^{-2/3}y^{2/3}}{2x^{1/3}y^{-1/3}}. \] Now, combine the exponents: - For \(x\): \(x^{-2/3}/x^{1/3} = x^{-2/3-1/3} = x^{-1}\), - For \(y\): \(y^{2/3}/y^{-1/3} = y^{2/3+1/3} = y\). Thus, we have \[ \frac{dy}{dx} = -\frac{y}{2x}. \] Evaluating at the point \((x,y) = (64,27)\): \[ \frac{dy}{dx}\bigg|_{(64,27)} = -\frac{27}{2(64)} = -\frac{27}{128}. \] **b. Interpretation of the result:** The derivative \[ \frac{dy}{dx} = -\frac{27}{128} \] at the point \((64,27)\) represents the rate at which capital spending (\(y\), in thousands of dollars) must change with respect to a change in labor spending (\(x\), in thousands of dollars) in order to keep the production level constant. A value of \(-\frac{27}{128}\) means that if the manufacturer increases labor spending by one thousand dollars, capital spending must decrease by approximately \(\frac{27}{128}\) thousand dollars (about \$210.94) in order to maintain the same level of production. This negative relationship indicates a trade-off between spending on labor and spending on capital along a production isoquant.