CalcAB Related Rates

$$A = \pi r^2$$ where A is the area and r is the radius.

The rate of change of area with respect to time is given by $$\frac{dA}{dt} = 2\pi r \frac{dr}{dt}$$ .

$$\frac{dr}{dt}$$ represents the rate at which the radius is changing with respect to time.

It means that the area is shrinking, and this rate can be expressed as $$\frac{dA}{dt} = -36\pi \frac{in^2}{hr}$$ .

$$V = \frac{4}{3}\pi r^3$$ where V is the volume and r is the radius.

Use the formula $$\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}$$ .

The negative sign indicates that the area is decreasing over time.

You can rearrange the area formula to find $$\frac{dr}{dt} = \frac{\frac{dA}{dt}}{2\pi r}$$ .

$$\frac{dV}{dt} = 32\pi \frac{in^3}{sec}$$ .

The rate of change of volume with respect to time is given by $$\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}$$ .