Under the given assumptions, summer lasts 93 days (10 days for Jun, 31 days each for Jul and Aug, and 21 days for Sept), and winter lasts 90 days (11 days for Dec, 31 days for Jan, 28 days for Feb, and 20 days for Mar).
1. In the summer, we assume the Earth moves in a circular path with radius 94.5 million mi. In 365 days, the planet would cover a distance equal to the circumference of this circle, 2π * (94.5 million mi), or approximately (using π = 3.14) 593.46 million mi. Summer last 93 days, which is roughly 93/365 = 25.48% of one year. So during the summer, the Earth will traverse about 25.48% of the circumference of its circular path, or
0.2548 * (593.46 million mi) = 151.2 million mi
In the winter, the Earth's path is taken to be a circle of radius 91.4 million mi, so its circumference is 2π * (91.4 million mi) = 573.99 million mi. Winter has a duration of 90/365 = 24.68% of a year, so during winter the Earth travels
0.2468 * (91.4 million mi) = 22.6 million mi
2. The average speed of a body over some time interval is the distance the body travels divided by the duration of time.
During the summer, the Earth moves at an average speed of
(151.2 million mi) / (93 days) = 1.63 million mi / day
During the winter, it moves at a speed of
(22.6 million mi) / (90 days) = 0.251 million mi / day
3. Use the distances/arc lengths found in (1) to determine the measure of the central angle θ subtended by the summer and winter arcs. In either case, the ratio of arc length to circumference is equal to the ratio between the central angle and one complete revolution (2π radians).
By the same token, the ratio of sector area A to the area of the whole circle (πr^2) is proportional to the ratio of the central angle to 2π rad.
Summer:
(151.2 million mi) / (2π * (94.5 million mi)) = θ / (2π rad)
==> θ = 1.6 rad
A / (π * (94.5 million mi)^2) = (1.6 rad) / (2π rad)
==> A = 7144.2 million square mi
Winter:
(22.6 million mi) / (2π * (91.4 million mi)) = θ / (2π rad)
==> θ = 0.2473 rad
A / (π * (91.4 million mi)^2) = (0.2473 rad) / (2π rad)
==> A = 1033.0 million square mi
