QPaperGen

The rate at which the height of sugar inside the cylindrical tank increases can be determined using the formula for the volume of a cylinder:

\(V = \pi r^2 h\)

Given:

• Radius, \(r = 10 \text{ cm}\)


• Rate of change of volume, \(\dfrac{dV}{dt} = 100\pi \text{ cm}^3/\text{s}\)

Since the radius remains constant, differentiate both sides of the volume equation with respect to time \(t\):

\(\dfrac{dV}{dt} = \pi r^2 \dfrac{dh}{dt}\)

Substitute the known values:

\(100\pi = \pi (10)^2 \dfrac{dh}{dt}\)

\(100\pi = 100\pi \dfrac{dh}{dt}\)

Dividing both sides by \(100\pi\):

\(\dfrac{dh}{dt} = 1 \text{ cm/s}\)

Therefore, the height of the sugar in the tank is increasing at a rate of \(1 \text{ cm/s}\).