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202122: applications, Rolle's theorem, grad. EVE Online is a space-based multi- player game, where players use spaceships to engage in combat, industry, exploration and other activities. a) A player is seeking to travel from her current position in a combat arena to the exit gate. The gate is located inside a cloud of gas, which is bounded by a plane: inside this gas, the ship's speed increases from v₁ to v₂. The diagram below explains the situation: she will travel from point A to point C by way of point B. (You may assume that any changes in speed or direction are instantaneous.)

202122: applications, Rolle's theorem, grad. EVE Online is a space-based multi- player game, where players use spaceships to engage in combat, industry, exploration and other activities. a) A player is seeking to travel from her current position in a combat arena to the exit gate. The gate is located inside a cloud of gas, which is bounded by a plane: inside this gas, the ship's speed increases from v₁ to v₂. The diagram below explains the situation: she will travel from point A to point C by way of point B. (You may assume that any changes in speed or direction are instantaneous.)

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202122: applications, Rolle's theorem, grad. EVE Online is a space-based multi- player game, where players use spaceships to engage in combat, industry, exploration and other activities. a) A player is seeking to travel from her current position in a combat arena to the exit gate. The gate is located inside a cloud of gas, which is bounded by a plane: inside this gas, the ship's speed increases from v₁ to v₂. The diagram below explains the situation: she will travel from point A to point C by way of point B. (You may assume that any changes in speed or direction are instantaneous.) с 02 T = 01 X - X X i. Explain why the time taken for the player to get to the exit gate is given by Y-y x² + y² √(x − x)² + (Y− y)² + V1 V2 ii. The player wishes to choose point B on the boundary in order to minimise the total travel time from point A to point C. Use calculus to show that, for this quickest trajectory, the angle 02 depends on 0₁ according to the relation sin 01 V1 sin 02 V2
Transcribed Image Text:202122: applications, Rolle's theorem, grad. EVE Online is a space-based multi- player game, where players use spaceships to engage in combat, industry, exploration and other activities. a) A player is seeking to travel from her current position in a combat arena to the exit gate. The gate is located inside a cloud of gas, which is bounded by a plane: inside this gas, the ship's speed increases from v₁ to v₂. The diagram below explains the situation: she will travel from point A to point C by way of point B. (You may assume that any changes in speed or direction are instantaneous.) с 02 T = 01 X - X X i. Explain why the time taken for the player to get to the exit gate is given by Y-y x² + y² √(x − x)² + (Y− y)² + V1 V2 ii. The player wishes to choose point B on the boundary in order to minimise the total travel time from point A to point C. Use calculus to show that, for this quickest trajectory, the angle 02 depends on 0₁ according to the relation sin 01 V1 sin 02 V2
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