## I made a mistake in the calculation. Let's fix it.

To find the rate of change of the base diameter r, we can use the formula:

dr/dt = (-0.5)(r/h)(dh/dt)

Given that dh/dt = 0.1 mm/s and h = 12.1 cm, we need to find the value of r.

To find r, we can use the equation for the volume of the cylinder:

V = Ï€r^2h

In this case, the volume V is constant, so we can write:

V = Ï€r^2h = Ï€(1 cm)^2(10 cm)

Simplifying, we have:

Ï€(1 cm)^2(10 cm) = Ï€r^2(12.1 cm)

Dividing both sides by Ï€ and 12.1 cm, we get:

r^2 = (1 cm^2 Ã— 10 cm) / 12.1 cm â‰ˆ 0.826 cm^2

Taking the square root of both sides, we find:

r â‰ˆ 0.909 cm

Now we can substitute the given values into the formula for dr/dt:

dr/dt = (-0.5)(0.909 cm / 12.1 cm)(0.1 mm/s)

Simplifying, we have:

dr/dt â‰ˆ -0.0376 mm/s

Therefore, the rate of change of the base diameter r when h = 12.1 cm is approximately -0.0376 mm/s.