Final answer:
To find the length of the curve, we can use the arc length formula and integrate the magnitude of the derivative of the position vector over the given range.
Step-by-step explanation:
To find the length of the curve, we can use the arc length formula:
L = ∫|r'(t)| dt
where r(t) is the position vector and r'(t) is the derivative of r(t) with respect to t. In this case, r(t) = cos(3t)i + sin(3t)j + 3 ln(cos(t))k. Taking the derivative of r(t), we get r'(t) = -3sin(3t)i + 3cos(3t)j - 3tan(t)ln(cos(t))i - 3ln(cos(t))sin(t)j. Now, we can find the magnitude of r'(t) which will give us the length of the curve:
|r'(t)| = sqrt((-3sin(3t))^2 + (3cos(3t))^2 + (-3tan(t)ln(cos(t)))^2 + (-3ln(cos(t))sin(t))^2). Integrate this expression over the given range 0 ≤ t ≤ π/4 to find the length of the curve.