Final answer:
The standard form of the equation with ( p = 6, φ = π/3 ) is not listed among the provided options. The proper form translates to r = 3 + 3*√3 or approximately r = 8.196 after converting to Cartesian coordinates.
Step-by-step explanation:
The question asks us to determine the standard form of the equation with parameters p = 6 and φ = π/3. When these parameters are applied to a polar equation of the form r = p, where r is the radius and φ is the angle in radians, we get the equation r(6) = 6. This equation equates to x = r*cos(φ) and y = r*sin(φ) in Cartesian coordinates, where x and y represent the horizontal and vertical axes, respectively. In this case, for φ = π/3, cos(π/3) is 1/2 and sin(π/3) is √3/2. Thus, our Cartesian coordinates become x = 6*(1/2) and y = 6*(√3/2), and plugging these values into the equation we obtain, 3 + 6*(√3/2) = r. The standard form of the equation is r = 3 + 6*(√3/2) which simplifies to r = 3 + 3√3 or approximately r = 8.196. None of the answer choices provided are correct since they do not represent this standard form.