# Math Help - Question on First order differential equation

1. ## Question on First order differential equation

Hi i have a question i think it is to do with first order differential equations.

The heat loss from a hot water tank is proportional to the temperature at any particular time. Twenty minutes after the water heater had been switched off, the temperature of the water in the tank was 50 degrees c
after forty minutes, the water temperature was 44 degrees c.

Determine
i) The temperature of the water when the water heater was switched off.
ii) The time taken for the water to cool to 40 degrees c.

2. Originally Posted by bmcciyem
Hi i have a question i think it is to do with first order differential equations.

The heat loss from a hot water tank is proportional to the temperature at any particular time. Twenty minutes after the water heater had been switched off, the temperature of the water in the tank was 50 degrees c
after forty minutes, the water temperature was 44 degrees c.

Determine
i) The temperature of the water when the water heater was switched off.
ii) The time taken for the water to cool to 40 degrees c.

$
\frac{dT}{dt} = kT$

$\frac{dT}{T} = kdt$

Integrate both sides

log(T) = kt + C

After t = 20mins T= 50

t= 40mins , T = 44

Use this two conditions for that equation and get the values of C and k

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i) put t= 0 for this & get T

ii) put T = 40 for this & get t