A point is moving along the graph of $$ x^{3} y^{2}=5488 $$. When the point is at $$ (7,4) $$, its $$ x $$-coordinate is changing at the rate of -6 units per minute. How fast is the $$ y $$-coordinate changing at that moment? The y-coordinate is changing by (Type an integer or a fraction. Simplif units. units per minute.

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To find how fast the $$ y $$-coordinate is changing, we can use implicit differentiation on the equation $$ x^3 y^2 = 5488 $$.

1. Differentiate both sides with respect to time $$ t $$:
   $$
   \frac{d}{dt}(x^3 y^2) = \frac{d}{dt}(5488)
   $$
   Since $$ 5488 $$ is a constant, its derivative is $$ 0 $$.

2. Apply the product rule to the left side:
   $$
   \frac{d}{dt}(x^3) \cdot y^2 + x^3 \cdot \frac{d}{dt}(y^2) = 0
   $$
   Using the chain rule, we have:
   $$
   3x^2 \frac{dx}{dt} y^2 + x^3 \cdot 2y \frac{dy}{dt} = 0
   $$

3. Rearranging gives:
   $$
   3x^2 y^2 \frac{dx}{dt} + 2x^3 y \frac{dy}{dt} = 0
   $$

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

5. Now, substitute $$ x = 7 $$, $$ y = 4 $$, and $$ \frac{dx}{dt} = -6 $$:
   $$
   \frac{dy}{dt} = -\frac{3(4)}{2(7)} (-6)
   $$
   $$
   \frac{dy}{dt} = -\frac{12}{14} (-6)
   $$
   $$
   \frac{dy}{dt} = \frac{12}{14} \cdot 6
   $$
   $$
   \frac{dy}{dt} = \frac{12 \cdot 6}{14} = \frac{72}{14} = \frac{36}{7}
   $$

Thus, the $$ y $$-coordinate is changing at a rate of $$ \frac{36}{7} $$ units per minute.

The final answer is:
$$
\frac{36}{7} \text{ units per minute.}
$$
The $$ y $$-coordinate is changing at a rate of $$ \frac{36}{7} $$ units per minute.

A point is moving along the graph of . When the point is at , its -coordinate is changing at the
rate of -6 units per minute. How fast is the -coordinate changing at that moment?
The y-coordinate is changing by
(Type an integer or a fraction. Simplif
units.
units per minute.

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A point is moving along the graph of . When the point is at , its -coordinate is changing at the rate of -6 units per minute. How fast is the -coordinate changing at that moment? The y-coordinate is changing by (Type an integer or a fraction. Simplif units. units per minute.