Water flows onto a flat surface at a rate of 5cm3/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 \(V=volume\), \(r=radius\), \(A=Area\) and \(d = depth\).
\[ V=A \cdot d \\ \therefore \\ V=A \cdot 0.1cm \\ \& \\ \frac{\partial}{\partial t}= 0.1cm \frac{\partial A}{\partial t} \] We know that \(\frac{\partial V}{\partial t} = 5cm \quad \therefore \quad 5=0.1 \frac{\partial A}{\partial t}\) therefor \(\frac{\partial A}{\partial t} = 50\)
\[ A=\pi r^2 \\ \frac{\partial A}{\partial t}=\frac{\partial}{\partial t}(\pi r^2) \\ \frac{\partial A}{\partial t}=2 \pi r \frac{\partial r}{\partial t} \]
Since we know that \(\frac{\partial A}{\partial t} = 50\) then \(50 = 2 \pi r \frac{\partial r}{\partial t}\) Thus the rate of change is. \[ \frac{50}{ 2 \pi r } = \frac{\partial r}{\partial t} \]
Thus to answer our questions:
nums <- c(1,10,100)
ans <- function(r){
50/(2 * pi * r)
}
ans(nums)## [1] 7.95774715 0.79577472 0.07957747Are the respective rates of change.