Math Problem Statement
A water tank is in the the shape of cone with its vertex pointing towards the ground. The tank has a height of 20 meters and its radius at the top of the tank is 5 meters. Water is being pumped into the tank at a rate of 3m3/min, but due to a hole at the bottom of tank, water is also leaking out at a rate proportional the water’s depth. (a) Derive an expression for the rate of change (with respect to time) of the depth of water in the tank. (b) Assuming that the water tank is able to be completely filled under these conditions, what can be determined about the proportionality constant? (c) The hole in the bottom of the tank has widened and water is now flowing out of the tank at a rate proportional to the cube of the depth of the water. Find a function for the height of the water in the tank at a given time. Assume that the tank was initially empty. Page 1 of 2
Solution
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Math Problem Analysis
Mathematical Concepts
Calculus
Differential Equations
Geometry
Formulas
Volume of a cone
Rate of change formula
Theorems
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Suitable Grade Level
Advanced High School or College