Question

Water flows on to 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? 1cm? 10cm? 100cm?

Answer 1

`5/[2pi],5/[20pi],5/[200pi].` `cms^-1`

Explanation:

Volume of regular cylinder `V` = `pir^2h`,

Differentiating implicitly with respect to `t` [ time] using the product rule.

`d[uv]=vdu+udv`, where `u` and `v` are each functions of some other variable, in this example `t` [where `u=r^2` and `v=h]`

So `d/dt[V]`= `pi[hd/dt[r^2]+r^2d/dt[h]]`=`pi[2rhdr/dt+r^2dh/dt]`......`[1]`

We know `[dV]/dt=5`, from the question, we also know that the height `h` is constant at `1 cm` [10mm] and `[dh]/dt` must equal zero , [since there is no change in height with respect to time.]

So ......`[1]` can expressed, `5=pi[2r[dr]/dt+0]` which yields,

`[dr]/dt = 5/[2pir`, i.e the rate of change of the radius with respect to time, from which the answers given above are obtained.