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

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**a. Finding $$\frac{dy}{dx}$$ and Evaluating at $$(64,27)$$**

We start with the production function
$$
40 x^{\frac{1}{3}} y^{\frac{2}{3}} = 480.
$$
Divide both sides by 40:
$$
x^{\frac{1}{3}} y^{\frac{2}{3}} = 12.
$$

Differentiate both sides with respect to $$x$$ implicitly. Using the product rule on the left side, we have
$$
\frac{d}{dx} \left( x^{\frac{1}{3}} y^{\frac{2}{3}} \right) = \frac{d}{dx}(12) = 0.
$$
Differentiating term by term:
$$
\frac{1}{3} x^{-\frac{2}{3}} y^{\frac{2}{3}} + x^{\frac{1}{3}} \cdot \frac{2}{3} y^{-\frac{1}{3}} \frac{dy}{dx} = 0.
$$

Solve for $$\frac{dy}{dx}$$:

1. Isolate the term with $$\frac{dy}{dx}$$:
   $$
   x^{\frac{1}{3}} \cdot \frac{2}{3} y^{-\frac{1}{3}} \frac{dy}{dx} = -\frac{1}{3} x^{-\frac{2}{3}} y^{\frac{2}{3}}.
   $$

2. Multiply both sides by 3 to eliminate the fractions:
   $$
   2 x^{\frac{1}{3}} y^{-\frac{1}{3}} \frac{dy}{dx} = - x^{-\frac{2}{3}} y^{\frac{2}{3}}.
   $$

3. Solve for $$\frac{dy}{dx}$$:
   $$
   \frac{dy}{dx} = -\frac{1}{2} \cdot \frac{x^{-\frac{2}{3}} y^{\frac{2}{3}}}{x^{\frac{1}{3}} y^{-\frac{1}{3}}}.
   $$

4. Combine the exponents for $$x$$ and $$y$$:
   - For $$x$$: 
     $$
     \frac{x^{-\frac{2}{3}}}{x^{\frac{1}{3}}} = x^{-\frac{2}{3} - \frac{1}{3}} = x^{-1}.
     $$
   - For $$y$$:
     $$
     \frac{y^{\frac{2}{3}}}{y^{-\frac{1}{3}}} = y^{\frac{2}{3} + \frac{1}{3}} = y.
     $$
     
Thus,
$$
\frac{dy}{dx} = -\frac{1}{2} \cdot \frac{y}{x}.
$$

Now, evaluate at $$(x,y) = (64, 27)$$:
$$
\frac{dy}{dx} \bigg|_{(64,27)} = -\frac{1}{2} \cdot \frac{27}{64} = -\frac{27}{128}.
$$

---

**b. Interpretation of the Result**

The derivative $$\frac{dy}{dx} = -\frac{27}{128}$$ at the point $$(64,27)$$ represents the rate at which spending on capital (in thousands of dollars) must change with respect to a change in spending on labor (in thousands of dollars) to keep the number of cars produced constant.

In other words, if the manufacturer increases spending on labor slightly (by, say, $$\Delta x$$ thousand dollars), the spending on capital must decrease by approximately $$\frac{27}{128} \Delta x$$ thousand dollars in order to maintain the same production level. The negative sign indicates that an increase in labor is associated with a decrease in capital expenditure along the production isoquant.
**a. Finding $$\frac{dy}{dx}$$ and Evaluating at $$(64,27)$$** Given the production function: $$ 40 x^{\frac{1}{3}} y^{\frac{2}{3}} = 480 $$ Divide both sides by 40: $$ x^{\frac{1}{3}} y^{\frac{2}{3}} = 12 $$ Differentiate both sides with respect to $$x$$: $$ \frac{1}{3} x^{-\frac{2}{3}} y^{\frac{2}{3}} + x^{\frac{1}{3}} \cdot \frac{2}{3} y^{-\frac{1}{3}} \frac{dy}{dx} = 0 $$ Solve for $$\frac{dy}{dx}$$: $$ \frac{dy}{dx} = -\frac{1}{2} \cdot \frac{y}{x} $$ At $$(64,27)$$: $$ \frac{dy}{dx} = -\frac{1}{2} \cdot \frac{27}{64} = -\frac{27}{128} $$ **b. Interpretation** The derivative $$\frac{dy}{dx} = -\frac{27}{128}$$ at $$(64,27)$$ means that for every thousand dollars increase in labor spending, capital spending must decrease by approximately $$\frac{27}{128}$$ thousand dollars to keep the number of cars produced constant. The negative sign indicates an inverse relationship between labor and capital expenditures in maintaining production levels.

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