Final answer:
To calculate the length of one turn of the helix, find the circumference of its base circle and the height of one turn, then use the Pythagorean theorem to find the hypotenuse of the right-angled triangle formed by these two lengths.
Step-by-step explanation:
To find the length of one turn of the helix given by the vector equation r(t) = (1/4)cos(t)i + (1/4)sin(t)j + (63/16)tk, we can consider the properties of a helix. The projection of the helix onto the xy-plane is a circle with radius r, and the length of one turn in the xy-plane is its circumference, which is 2πr. Additionally, the helix advances by a step 'h' in the z-direction after one complete revolution. The step 'h' corresponds to the coefficient of the t term in the k component of the equation, which here is (63/16)2π. The length of one complete turn of the helix is the hypotenuse of a right-angled triangle with the circumference as one side and the step height as the other.
Therefore, the length L is given by L = √((2πr)^2 + h^2). Substituting the values, we get L = √((2π*1/4)^2 + (63/16*2π)^2). Computing this value gives us the length of one turn of the helix.