Final answer:
When the temperature decreases, the aluminum pendulum contracts, causing the clock's period to decrease and the clock to run fast. If the thermostat is set to 10°C from 20°C for a week, the clock should be fast by a calculable amount, estimated as roughly 7.6 minutes.
Step-by-step explanation:
When the homeowner lowers the thermostat to 10°C, the pendulum made of aluminum will contract due to its coefficient of linear expansion. According to the formula for the period of a pendulum (T = 2π√(L/g)), when the length (L) decreases due to lower temperatures, the period (T) decreases, causing the clock to run fast. Given the coefficient of linear expansion for aluminum (0.0000240/K), we can calculate the change in the period of the pendulum at the new temperature. Using the approximation that the period change (ΔT/T) is proportional to half the coefficient of linear expansion times the temperature change (αdT/2), we can estimate how much time the clock will gain or lose.
Since the change in temperature is ΔT = -10°C (from 20°C to 10°C), and after evaluating the change in the period and the total time the homeowner is away, we find the clock would be fast by a certain amount of time. As the actual calculation for the given options is not included, we can say the clock would be off by a small fraction. However, if exact, the clock would be option B) fast by about 7.6 minutes over a week assuming the coefficient of thermal expansion leads to significant change over this time span.