A right circular conical tank with its tip downward has a height of 6 m and a base…
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
A right circular conical tank with its tip downward has a height of 6 m and a base radius of 4 m. Water is being pumped into the tank at a constant rate of 2 m3/min while simultaneously leaking through a hole in the tip at a rate of 1 m3/min.
Let h(t) denote the depth (in meters) of the water in the tank at time t minutes and let r(t) denote the radius of the water’s surface at that time. Assume the water surface is always a circle and the water exactly fits the shape of the cone at every instant.
(a) Express r as a function of h by relating the geometry of the water surface to the dimensions of the tank.
(b) Derive an equation that relates the volume V of water in the tank to the water depth h. Do not differentiate yet.
(c) Differentiate your equation implicitly with respect to time t to obtain an expression for dtdh in terms of h and dtdV.
(d) Using the given inflow and outflow rates, determine the rate of change of the water depth, dtdh, at the instant when the water is 3 m deep. Include appropriate units.
(e) Interpret the sign of dtdh found in part (d) within the physical context of the tank.