Final answer:
The parametric equations for the tangent line are x(t) = cos(56π) + 56πt, y(t) = sin(56π) + t, and z(t) = 56πt, where the direction of the tangent is parallel to (56π, 1, 56π).
Step-by-step explanation:
The correct parametric equations for the tangent line at the point (cos(56π), sin(56π), 56π) on the curve are x(t) = cos(56π) + 56πt, y(t) = sin(56π) + t, z(t) = 56πt. This means the tangent has a slope parallel to the vector (56π, 1, 56π), which represents the direction of the tangent line.To find the tangent line, you need to know a point on the line and a direction vector. The given point is (cos(56π), sin(56π), 56π). Using the principle that the slope of the curve at a given point is equal to the slope of the tangent at that point, we find that the direction of the tangent vector at the curve can be represented by the derivatives of each parametric equation with respect to 't'. The derivatives are the coefficients of 't' in the equations.
The parametric equations for the tangent line at the point (cos(56π), sin(56π), 56π) on the curve are:x(t) = cos(56π) + ty(t) = sin(56π) + 56πtz(t) = 56π + tExplanation: To find the parametric equations for the tangent line, we need to find the slope of the curve at the given point. The slope can be found by taking the derivative of each component of the curve with respect to t. Then, substitute the given values into the equation to get the parametric equations for the tangent line.