The volume of a cube is increasing at a rate of 10 cm^3/min. How fast is the surface area increasing when the length of an edge 90 cm?

1 Answer
Barry H.
May 24, 2018

#12/27# cms squared per minute.

Explanation:

Let the length of a side of the cube =#x#

Then volume of cube # v # =#x^3#.........#[1]#
Surface area of cube #s# will = #6x^2#......#[2]#

Differentiating #[1]# implicitly w.r.t. #t#[ time], #[dv]/dt=3x^2dx/dt# , but we know from the question that #[dv]/dt=10#, therefore #dx/dt=[10]/[3x^2#......#[3]#

Differentiating.......#[2]# w.r.t. #t#, #[ds]/dt=12xdx/dt# and substituting for #dx/dt# from .....#[3]# we obtain,

#[ds]/dt=[12x[10]/[3x^2]]# = #[120]/[3x]#, and so we have the rate of change of the side length of the cube with respect to time, and when #x=90#, #[ds]/dt=120/[3[90]]# = #12/27#.

Hope this was helpful.

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