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Math Help - Wordy question, about rate of temperature increase.

  1. #1
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    Wordy question, about rate of temperature increase.

    A tank contains water which is heated by an electric water heater working under the the action of a thermostat. The temperature of the water, \theta degrees C, may be modelled as follows. When the water heater is first switched on, \theta=40. The heater causes the temperature to increase at a rate K1 degrees C per second, where K1 is a constant, untill \theta=60. The heater then switches off.

    i)Write down, in terms of K1, how long iit takes for the temperature to increase from 40degreesC to 60degreesC.


    The temperature of the water then immediately starts to decrease at a variable rate K2(\theta-20)degreesC per second, where K2 is a constant, untill \theta=40.

    ii)Write down a differential equation to represent the situation as the temperature is decreasing.

    iii)Fi nd the total length of time for the temperature to increase from 40degreesC to 60degreesc and then decrease to 40degreesC. Give your answer in terms of K1 and K2

    Long question has confused me. Please show me how to do it.
    Thanks
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    Quote Originally Posted by George321 View Post
    A tank contains water which is heated by an electric water heater working under the the action of a thermostat. The temperature of the water, \theta degrees C, may be modelled as follows. When the water heater is first switched on, \theta=40. The heater causes the temperature to increase at a rate K1 degrees C per second, where K1 is a constant, untill \theta=60. The heater then switches off.

    i)Write down, in terms of K1, how long iit takes for the temperature to increase from 40degreesC to 60degreesC.


    The temperature of the water then immediately starts to decrease at a variable rate K2(\theta-20)degreesC per second, where K2 is a constant, untill \theta=40.

    ii)Write down a differential equation to represent the situation as the temperature is decreasing.

    iii)Fi nd the total length of time for the temperature to increase from 40degreesC to 60degreesc and then decrease to 40degreesC. Give your answer in terms of K1 and K2

    Long question has confused me. Please show me how to do it.
    Thanks
    (i) 60 = 40 + k_1t ... solve for t

    (ii)  \frac{d\theta}{dt} = k_2(\theta - 20)

    (iii) total time = t from part (1) + t for cool down

    solve the DE for \theta as a function of time using the initial value \theta(0) = 60 ... then determine the time for cool down to 40.
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