U3P2 - Related Rates

The volume of a sphere is given by the formula: $$V = \frac{4}{3} \pi r^3$$ .

A related rates problem involves finding the rate at which one quantity changes with respect to another quantity that is also changing.

Use the chain rule: $$\frac{dV}{dt} = \frac{dV}{dr} \cdot \frac{dr}{dt}$$ .

The derivative is $$\frac{dV}{dr} = 4\pi r^2$$ .

This is expressed as $$\frac{dr}{dt} = 2 \text{ cm/sec}$$ .

The relationship is given by the Pythagorean theorem: $$L^2 = h^2 + b^2$$ , where L is the length of the ladder, h is the height, and b is the base.

Differentiate the Pythagorean theorem with respect to time to find $$\frac{db}{dt}$$ .

The area A of a rectangle is given by $$A = l \times w$$ , where l is the length and w is the width.

Use the product rule: $$\frac{dA}{dt} = l \frac{dw}{dt} + w \frac{dl}{dt}$$ .

An increasing quantity has a positive rate of change, while a decreasing quantity has a negative rate of change.