Final answer:
The angle of intersection between the given curves is approximately 64 degrees. The correct answer is (option a).
Step-by-step explanation:
The angle of intersection between two curves can be found by taking the dot product of their tangent vectors and dividing it by the product of their magnitudes. The tangent vectors can be found by taking the derivative of the position vectors with respect to the parameter t. So, for the curves r₁(t)=(3t,t²,t⁴) and r₂(t)=(sin(t),sin(3t),4t), we differentiate them with respect to t to get their tangent vectors r₁'(t)=(3,2t,4t³) and r₂'(t)=(cos(t),3cos(3t),4).
Next, we calculate the dot product of the tangent vectors: r₁'(t) · r₂'(t) = 3cos(t)+6tcos(3t)+16t³. Then, we find the magnitudes of the tangent vectors: |r₁'(t)| = √(9+4t²+16t⁶) and |r₂'(t)| = √(1+9cos²(3t)+16). Lastly, we divide the dot product by the product of the magnitudes and take the inverse cosine to find the angle of intersection, which turns out to be approximately 64 degrees (option a).