Question

Water is falling on a surface, wetting a circular area that is expanding at a rate of 4 mm2 /s. How fast is the radius of the wetted area expanding when the radius is 150 mm

Answer 1

Answer:

The radius of the wetter area expands at a rate of $4.244\times 10^{-3}$ milimeters per second when radius is 150 milimeters.

Explanation:

From Geometry we remember that area of a circle is described by this expression:

$A =\pi\cdot r^{2}$ (Eq. 1)

Where:

$r$ - Radius of the circle, measured in milimeters.

$A$ - Area of the circle, measured in square milimeters.

Then, the rate of change of the area in time is derived by concept of rate of change, that is:

$\frac{dA}{dt} = 2\pi\cdot r\cdot \frac{dr}{dt}$ (Eq. 2)

Where:

$\frac{dr}{dt}$ - Rate of change of radius in time, measured in milimeters per second.

$\frac{dA}{dt}$ - Rate of change of area in time, measured in square milimeters per second.

Now the rate of change of radius in time is cleared within equation above:

$\frac{dr}{dt} = \left(\frac{1}{2\pi\cdot r}\right)\cdot \frac{dA}{dt}$

If we know that $r = 150\,mm$ and $\frac{dA}{dt} = 4\,\frac{mm^{2}}{s}$, then the rate of change of radius in time is:

$\frac{dr}{dt} = \left[\frac{1}{2\pi\cdot (150\,m)} \right] \cdot \left(4\,\frac{mm^{2}}{s} \right)$

$\frac{dr}{dt}\approx 4.244\times 10^{-3}\,\frac{mm}{s}$

The radius of the wetter area expands at a rate of $4.244\times 10^{-3}$ milimeters per second when radius is 150 milimeters.