A conical (cone shaped) water tower has a height of 12 ft and a radius of 3 ft. Water is pumped into the tank at a rate of 4 ft^3/min. How fast is the water level rising when the water level is 6 ft?
Accepted Solution
A:
Answer:The water rises at a rate of 16/(9π) ft/min or approximately 0.566 ft/min.Step-by-step explanation:Volume of a cone is:V = ⅓ π r² hUsing similar triangles, we can relate the radius and height of the water to the radius and height of the tank.r / h = R / Hr / h = 3 / 12r = ¼ hSubstitute:V = ⅓ π (¼ h)² hV = ¹/₄₈ π h³Take derivative with respect to time:dV/dt = ¹/₁₆ π h² dh/dtPlug in values:4 = ¹/₁₆ π (6)² dh/dtdh/dt = 16 / (9π)dh/dt ≈ 0.566The water rises at a rate of 16/(9π) ft/min or approximately 0.566 ft/min.