EVE Online is a space-based multiplayer game, where players use spaceships to engage in combat, industry, exploration and other activities. Players can design their own ships, incorporating ship systems that affect a range of different critical variables. In combat situations, one such critical variable is ”capacitor recharge rate”: the capacitor stores energy used for activating both offensive and defensive ship systems. A spaceship that runs out of energy rarely survives for long. The capacitor recharge rate varies with how full the capacitor is currently. The recharge rate is determined by the differential equation dC/dt = 10 M/T [√(C/M) − C/M] where M is the capacitor’s overall capacity for holding energy (measured in gigajoules, GJ), and T is the time (measured in seconds, s) taken to recharge from zero to a standardised fraction of M. a) Using Rolle’s theorem, infer that there is a maximum value for dC/dt for some C ∈ (0, M). b) Treating M and T as constant, show that dC/dt is at a maximum when C = 0.25M. c) The Punisher-class frigate has a basic capacitor capacity of 460 GJ, with a recharge time of 136 s. Use calculus to estimate which will have the greater impact on peak recharge rate: an increase in capacity of 3 GJ, or a decrease in recharge time of 1 s.
EVE Online is a space-based multiplayer game, where players use spaceships to engage in combat, industry, exploration and other activities. Players can design their own ships, incorporating ship systems that affect a range of different critical variables. In combat situations, one such critical variable is ”capacitor recharge rate”: the capacitor stores energy used for activating both offensive and defensive ship systems. A spaceship that runs out of energy rarely survives for long. The capacitor recharge rate varies with how full the capacitor is currently. The recharge rate is determined by the differential equation
dC/dt = 10 M/T [√(C/M) − C/M]
where M is the capacitor’s overall capacity for holding energy (measured in gigajoules, GJ), and T is the time (measured in seconds, s) taken to recharge from zero to a standardised fraction of M.
a) Using Rolle’s theorem, infer that there is a maximum value for dC/dt for some C ∈ (0, M).
b) Treating M and T as constant, show that dC/dt is at a maximum when C = 0.25M.
c) The Punisher-class frigate has a basic capacitor capacity of 460 GJ, with a recharge time of 136 s. Use calculus to estimate which will have the greater impact on peak recharge rate: an increase in capacity of 3 GJ, or a decrease in recharge time of 1 s.
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