## To find the distance from Earth to Venus for both positions of Venus, we can use the law of cosines. The law of cosines states that in a triangle with sides a, b, and c and angle C opposite side c, the following equation holds true:

c^2 = a^2 + b^2 - 2ab * cos(C)

In this case, the distance from Earth to Venus can be viewed as the side c of a triangle, with the radii of Earth's orbit and Venus's orbit as sides a and b respectively. The angle between these sides, in this case, is the difference between the positions of Venus.

Given that the radius of Earth's orbit is 1.5 * 10^8 km and the radius of Venus's orbit is 1.1 * 10^8 km, we can calculate the distance from Earth to Venus for the first position as follows:

c^2 = (1.5 * 10^8)^2 + (1.1 * 10^8)^2 - 2 * (1.5 * 10^8) * (1.1 * 10^8) * cos(30)

By evaluating this equation, we find that the distance from Earth to Venus for the first position is approximately 2.1 * 10^8 km.

Similarly, for the second position, we can calculate:

c^2 = (1.5 * 10^8)^2 + (1.1 * 10^8)^2 - 2 * (1.5 * 10^8) * (1.1 * 10^8) * cos(150)

By evaluating this equation, we find that the distance from Earth to Venus for the second position is approximately 0.49 * 10^8 km.

Therefore, the corrected distances from Earth to Venus for the given positions are approximately 2.1 * 10^8 km and 0.49 * 10^8 km.