A storage vessel in the shape of a right circular cone has a height of 12 m and a…
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A storage vessel in the shape of a right circular cone has a height of 12 m and a base radius of 4 m. Water is being pumped into the vessel, and the depth of the water at a particular instant is h meters.
At that instant the pump is running but no numerical flow rate is provided.
Answer the following questions about the geometric and analytic relationships that will later be used to model the changing water level.
(a) Use geometric reasoning to obtain an explicit algebraic expression for the radius r of the water’s surface in terms of the depth h. Clearly state any proportional relationships you employ.
(b) Write an expression for the volume V of water in the vessel as an explicit function of h using the result from part (a). Include correct units in your final formula.
(c) Describe, in complete sentences, how differentiating the expression from part (b) with respect to time t would relate dtdV and dtdh, and explain why such a relationship is useful for analyzing how quickly the water level changes while the pump is running.