Math Problem Statement

Solution

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Math Problem Analysis

Mathematical Concepts

Calculus
Volume Integrals
Work-Energy Principle
Geometry of Solids

Formulas

Work = ∫ Force × Distance
Volume of a cone slice: V = πr(y)^2 dy
Volume of a sphere slice: V = π(64 - y²) dy
Distance to pump water = height - current y position

Theorems

Work-Energy Theorem
Similar Triangles (for cone)
Pythagorean Theorem (for sphere)

Suitable Grade Level

College level calculus or advanced high school