Suppose That $z = x ^ { 3 } y ^ { 2 }$

question 191

Essay

Suppose that $z = x ^ { 3 } y ^ { 2 }$ , where both $x$ and $y$ are changing with time. At a certain instant when $x = 1$ and $y = 2$ , $x$ is decreasing at the rate of 2 units/s and $y$ is increasing at the rate of 3 units/s. How fast is $Z$ changing at this instant? Is $Z$ increasing or decreasing?

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Definitions:

Ozone Destroyer

Substances, typically man-made chemicals like CFCs, that lead to the depletion of the ozone layer in the Earth's stratosphere.

Deforestation

The large-scale removal of forest land or tree plantations, often resulting in damage to the quality of the land.

Soil Nutrient Loss

The depletion of essential minerals and compounds from soil, often due to factors like erosion, over-farming, or leaching, which can reduce soil fertility and agricultural productivity.

Global Climate Change

An ongoing trend toward warmer temperatures and shifts in climate patterns worldwide; caused by a rise in greenhouse gases.