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?

1 Answer
May 25, 2018

#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.