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
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