Respuesta :
The temperature of a substance will naturally cool down over time due to heat transfer to the surrounding environment. The rate at which the substance cools down is described by Newton's law of cooling, which states that the rate of change of the temperature of a substance is proportional to the difference in temperature between the substance and the surroundings. Mathematically, this can be expressed as:
dT/dt = k(T - Ts)
Where T is the temperature of the substance, Ts is the temperature of the surroundings, t is time, and k is the cooling coefficient, which is a constant that depends on the properties of the substance and the surroundings.
Since we know that the initial temperature of the coffee is 180 and the temperature of the surroundings is 80, and the temperature of the coffee after 10 minutes is 130, we can write the following equation:
(180 - 130)/10 = k(180 - 80)
Solving for k, we get:
k = -4
Substituting this value back into the equation for the rate of change of temperature, we get:
dT/dt = -4(T - 80)
This is the equation for the temperature of the coffee as a function of time. To find the temperature of the coffee at any time t, we can integrate this equation with respect to time. The resulting equation will be:
[tex]T = 80 + C*e^{(-4t)[/tex]
Where C is an unknown constant of integration. Since we are given that the initial temperature of the coffee is 180, we can solve for C by substituting t = 0 and T = 180 into the equation:
[tex]180 = 80 + C*e^0[/tex]
Solving for C, we get:
C = 100
Substituting this value back into the equation for T, we get the final equation for the temperature of the coffee as a function of time:
[tex]T = 80 + 100*e^{(-4t)[/tex]
This is the final, simplified equation for the temperature of the coffee as a function of time. There are no unknown constants in this equation.
To learn more about temperature, visit:
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