Many home heating systems operate by pumping hot water through radiator pipes. The flow of the water to different "zones" in the house is controlled by zone valves that open in response to thermostats. The opening and closing of a zone valve is commonly performed by a wax actuator, as shown in the diagram. When the thermostat signals the valve to open, a dc voltage of 24 V is applied across a heating element (resistance R = 200 Ω) in the actuator. As the wax melts, it expands and pushes a cylindrical rod (radius 2.0 mm) out a distance 1.0 cm to open the zone switch. The actuator contains 2.0 mL of solid wax of density 0.90 g/cm3 at room temperature (20°C). The specific heat of the wax is 0.80 J/(g·°C), its latent heat of fusion is 60 J/g, and its melting point is 90°C. When the wax melts its volume expands by 15%. How long does it take until the valve is fully open?

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Wax Heating element Cylindnical rod to Thermostat Rubber diaphragm Wax actuator