Industrial lubricant is stored in a container shaped like a right frustum of a cone
MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation
Problem
Industrial lubricant is stored in a container shaped like a right frustum of a cone.
• The container height is 10 m, the radius at the bottom is 1 m, and the radius at the top is 3 m.
• Lubricant is pumped into the container at 0.20 m3 min−1 while simultaneously leaking from a small crack at 0.05 m3 min−1.
• The lubricant’s temperature is rising at 0.30 °C min−1 due to ambient conditions, and its coefficient of volumetric thermal expansion is 9.5×10−4 °C−1.
Let h(t) represent the depth of lubricant (in meters) at time t minutes, and assume the lubricant is well-mixed so its temperature is spatially uniform.
(a) Derive a formula for the radius of the lubricant surface, r, as a linear function of its depth h. Briefly justify each step.
(b) Write an expression for the net rate of volume change, dtdV, that accounts for inflow, outflow, and thermal expansion. Express your result in terms of h and the lubricant’s instantaneous volume V.
(c) Obtain a formula for dtdh in terms of h by relating volume change to depth change. Then evaluate dtdh when h=6.0 m and the lubricant temperature is 20 °C.
(d) Suppose an engineer neglected thermal expansion when estimating dtdh. Would this omission cause the estimate to be too large, too small, or unbiased at h=6.0 m? Provide a quantitative or qualitative justification based on your work in part (b).