Related Rates of Change (1 Viewer)

[locked.on]

The following are two questions which I couldn't obtain the solution provided.
It may be a mistake I haven't picked up in my working or an error after all.

Any worked answers or solutions only are much appreciated.

Thanks in advance.

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Q1

The volume of a spherical ball, of radius 24mm, decreases at a rate equal to three times its surface area at that time.

Find the time taken for the volume to be reduced to one-eighth of its original size.

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Q2

The radius of a cylinder increases at a constant rate of 0.1cm per minute and its height remains constant.

At what rate is the volume of the cylinder, whose height is 10cm, increasing when the radius is 2cm?

 

[azureus88]

1. dV/dt = (dV/dr)(dr/dt)
12πr^2 = (4πr^2)(dr/dt)
dr/dt = 3

ok, this is the (sort of) tricky part:

We assume that the sphere retains its shape when it changes volume so
the time taken for volume to be reduced by factor of 1/8( = 1/2^3) is actually equal to the time taken for radius to shrink by factor of 1/2.

so we want to know how long it takes to reduce from 24 to 12 and since dr/dt = 3, T = (24-12)/3 = 4 units of time

2. dV/dt = (dV/dr)(dr/dt)
=(2(pi)hr)0.1 sub in h=10, r=2 (note that h is constant)
=4pi