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 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.
**a. Finding $$\frac{dy}{dx}$$ and evaluating at $$(64,27)$$:** Given the equation: $$ 40x^{1/3}y^{2/3} = 480 $$ Differentiate both sides with respect to $$x$$: $$ \frac{dy}{dx} = -\frac{y}{2x} $$ At the point $$(64,27)$$: $$ \frac{dy}{dx} = -\frac{27}{128} $$ **b. Interpretation:** The negative value of $$\frac{dy}{dx} = -\frac{27}{128}$$ indicates that increasing labor spending by \$1,000 requires decreasing capital spending by approximately \$210.94 to maintain the same production level.

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