It takes light 8 minutes to reach the Earth from the Sun. Neptune is 30 times further away from the Sun than the Earth is. What’s the time delay in a conversation
with someone at Neptune when the planets are both at their closest and at their
furthest?

The time delay  in a conversation with someone at Neptune when the planets are at their closest is 240 minutes

The time delay  in a conversation with someone at Neptune when the planets are at their furthest is 256 minutes

Explanation:

The parameters given are as follows;

Time for duration of a signal to the Earth from the Sun = 8 minutes

Distance of the Earth from Neptune = 30 × Distance of the Earth from the Sun

Therefore, where distance from Neptune to the Sun = x

x = 30 × Distance of the Earth from the Sun + Distance of the Earth from the Sun

x = 31 × Distance of the Earth from the Sun

Closest distance from the Earth to Neptune = 30 × Distance of the Earth from the Sun

∴ Farthest distance from the Earth to Neptune = Distance from Neptune to the Sun + Distance of the Earth from the Sun

Farthest distance from the Earth to Neptune = x + Distance of the Earth from the Sun

Farthest distance from the Earth to Neptune = 32× Distance of the Earth from the Sun

Hence, the time in a conversation with someone at Neptune when the planets are at their closest is given s follows;

Closest distance from the Earth to Neptune = 30 × Distance of the Earth from the Sun

Therefore, time for duration of a signal to the Earth from Neptune (when both planets are closest) = 30 × Time for duration of a signal to the Earth from the Sun

Time delay of a signal to the Earth from Neptune (when both planets are closest) = 30 × 8 minutes = 240 minutes

Similarly;

Time delay of a signal to the Earth from Neptune (when both planets are furthest) = 32 × 8 minutes = 256 minutes.

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