Math Problem Statement
Sand is poured at a constant rate onto a flat horizontal surface. It forms a pile in the shape of a cone with a constant slope.
[3 marks] If r is the radius of the base of the cone, find how r varies with t, given that r (0) = 0.
[2 marks] If it takes a time t1 for r to reach r1, how long does it take for the radius to reach 2r1 (in terms of r1)?
Solution
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Math Problem Analysis
Mathematical Concepts
Calculus
Geometry
Formulas
Volume of a cone: V = (1/3)πr²h
Constant slope: h = kr
Volume proportionality: V = C r³
Rate of change of volume: dV/dt = Q
Radius variation: r(t) = (Qt/C)^(1/3)
Theorems
Fundamental theorem of calculus
Chain rule in differentiation
Suitable Grade Level
Undergraduate Calculus
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