Water flows on to a flat surface at a rate of \(5cm^3/s\) forming a circular puddle \(10mm\) deep. How fast is the radius growing when the radius is
a)1 cm?
b)10 cm?
c)100 cm?
Let V = volume, r = radius, A = area, d = depth \[V= A.d\] \[V=A*0.1\] \[\frac{\partial }{\partial x}=0.1\frac{\partial A}{\partial x} \]
Given \(5 cm =\frac{\partial V}{\partial x}\) \[\therefore \frac{\partial A}{\partial x}=50 \]
\[A=\pi r^2 \] \[\frac{\partial A}{\partial x}=\frac{\partial }{\partial x}(\pi r^2) \] \[\frac{\partial A}{\partial x}=2\pi r\frac{\partial r}{\partial x} \]
Substituting \(\frac{\partial A}{\partial x}\) \[\frac{\partial r}{\partial x}=\frac{50}{2\pi r}\]
# implement above function to find the change of rate of the radius
a <- function(x){50/(2 * pi * x)}
a(c(1, 10, 100))## [1] 7.95774715 0.79577472 0.07957747