Math Problem Statement

Consider the path r(t) = (16t, 8t^2, 8ln(t)) defined for t > 0. Find the length of the curve between the points (16, 8, 0) and (80, 200, 8ln(5)).

Solution

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Math Problem Analysis

Mathematical Concepts

Vector Calculus
Parametric Curves
Arc Length

Formulas

Arc length formula: L = ∫ |r'(t)| dt from t=a to t=b
r(t) = (16t, 8t^2, 8ln(t))
r'(t) = (16, 16t, 8/t)
|r'(t)| = sqrt(16^2 + (16t)^2 + (8/t)^2)

Theorems

Arc Length Formula for Parametric Curves

Suitable Grade Level

College/University Calculus

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