A cylindrical bucket is being filled with paint at a rate of 6 cm per minute. How fast is the level rising when the bucket starts to overflow? The bucket has a radius of 30 cm and a height of 60 cm.

Q: A cylindrical bucket is being filled with paint at a rate of 6 cubic cm per minute. How fast is the level rising when the bucket starts to overflow? The bucket has a radius of 30 cm and a height of 60 cm.

Answer:

0.00212 cm/min

Step-by-step explanation:

Volume of a cylinder

V = πr²h ... Equation 1

Where r = radius of the cylinder, h = height of the cylinder, π = pie.

From Chain Rule

dV/dt = (dV/dh) * (dh/dt) ... Equation 2

Differentiating equation 1 with respect to h

dVdh = πr² ... Equation 3

Given: r = 30 cm, π = 3.14

Substitute into equation 3

dV/dh = 3.14 (30) ²

dV/dh = 2826 cm²

But, dV/dt = 6 cm³/min.

Also substituting into equation 2

6 (cm³/min) = 2826 (cm²) * dh/dt

Making dh/dt the subject of the equation

dh/dt = 6 (cm³/min) / 2826 (cm²)

dh/dt = 0.00212 cm/min.

Thus the level is rising at 0.00212 cm/min