Final answer:
To calculate the radius of Venus's orbit in Astronomical Units, the angular separation is used in conjunction with trigonometry. The cosine function gives a ratio that, multiplied by 1 AU, provides the radius, yielding an approximate value of 0.694 AU.
Step-by-step explanation:
To calculate the radius of Venus's orbit in Astronomical Units (AU), we consider the largest angular separation between Venus and the Sun when viewed from Earth, which is 46 degrees.
This separation occurs when Venus is at its greatest elongation.
Since we are approximating that Venus's orbit is circular, we can use simple geometry to relate the angular separation to the radius of the orbit.
The angular separation of 46 degrees is nearly a right angle, so we expect Venus to be at a position in its orbit where the Earth, Venus, and the Sun approximately form a right triangle, with Venus at the vertex opposite the right angle. The distance from Venus to the Sun is the adjacent side of the triangle, and the distance from Earth to the Sun (1 AU) is the hypotenuse.
Using trigonometry, specifically the cosine function, we can calculate the distance from Venus to the Sun.
The cosine of the angle gives us the ratio of the adjacent side to the hypotenuse:
cos(46 degrees) = radius of Venus's orbit / 1 AU
The radius of Venus's orbit is then equal to cos(46 degrees) times 1 AU.
we find that cos(46 degrees) is approximately 0.694.
Therefore:
Radius of Venus's orbit ≈ 0.694 AU
Since the question defines the average distance of Venus from the Sun as 0.72 AU, our calculated value is reasonably close to this average distance.