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?
1 answer:
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 h
Using similar triangles, we can relate the radius and height of the water to the radius and height of the tank.
r / h = R / H
r / h = 3 / 12
r = ¼ h
Substitute:
V = ⅓ π (¼ h)² h
V = ¹/₄₈ π h³
Take derivative with respect to time:
dV/dt = ¹/₁₆ π h² dh/dt
Plug in values:
4 = ¹/₁₆ π (6)² dh/dt
dh/dt = 16 / (9π)
dh/dt ≈ 0.566
The water rises at a rate of 16/(9π) ft/min or approximately 0.566 ft/min.
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