Key mathematical problems 123 1. Xiao Ming is making a model of the earth with a diameter of D meters: (1) how long does it take for him to surround the equator with a wire? (2) how long does it take to surround the equator if he wants to increase the radius of the model by M meters? (3) how long does it take to estimate a circle of wire for the equator of the earth? If he wants to increase the radius of the circle of wire by M meters So how long wire need to be added (the radius of the earth is about 6370km) (4) compare the results of (2) and (3), what do you find? Why?

Key mathematical problems 123 1. Xiao Ming is making a model of the earth with a diameter of D meters: (1) how long does it take for him to surround the equator with a wire? (2) how long does it take to surround the equator if he wants to increase the radius of the model by M meters? (3) how long does it take to estimate a circle of wire for the equator of the earth? If he wants to increase the radius of the circle of wire by M meters So how long wire need to be added (the radius of the earth is about 6370km) (4) compare the results of (2) and (3), what do you find? Why?

Xiao Ming is making a model of the earth with a diameter of D meters: (1) if he wants a wire to encircle the equator, how long does it take? If the diameter is D and the radius is r = D / 2, then the perimeter is L = 2 π r = 2 π * (D / 2) = π D (2) if he wants to increase the radius of this model by M meters, how long does it take to encircle the equator? Increase the radius by M

In △ ABC, the opposite sides of ∠ a ·, ∠ B, and ∠ C are ABC, and ∠ A is an acute angle, and sin ˇ 2 (a) - cos ˇ 2 (a) = 1 / 2. It is proved that B + C ≤ 2A

Analysis: ∵ (Sina) ^ 2 - (COSA) ^ 2 = 1 / 2, ∵ (COSA) ^ 2 - (Sina) ^ 2 = - 1 / 2, that is, cos2a = - 1 / 2 ∵ 2A = 120 °, ∵ a = 60 ° cosa = cos60 ° = (b ^ 2 + C ^ 2-A ^ 2) / 2BC = 1 / 2 ∵ a ^ 2 = B ^ 2 + C ^ 2-bc and 3 (B-C) ^ 2 ≥ 0, that is, 3b ^ 2-6bc + 3C ^ 2 ≥ 04 (b ^ 2 + C ^ 2-bc) ≥ B ^ 2 + C ^ 2 + 2BC ∵ 4A ^ 2 ≥