The differential equation below models the temperature of a 95°C cup of coffee in a 21°C room, where it is known that the coffee cools at a rate of 1°C per minute when its temperature is 71°C. Solve the differential equation to find an expression for the temperature of the coffee at time t. (Let y be the temperature of the cup of coffee in °C, and let t be the time in minutes, with t = 0 corresponding to the time when the temperature was 95°C.) dydt = − 150(y − 21)
Question
The differential equation below models the temperature of a 95°C cup of coffee in a 21°C room, where it is known that the coffee cools at a rate of 1°C per minute when its temperature is 71°C. Solve the differential equation to find an expression for the temperature of the coffee at time t. (Let y be the temperature of the cup of coffee in °C, and let t be the time in minutes, with t = 0 corresponding to the time when the temperature was 95°C.) dydt = − 150(y − 21)
Solution
The given differential equation is a first order linear differential equation and it can be solved using the method of integrating factors.
The differential equation is given by:
dy/dt = -150(y - 21)
Rearranging the equation, we get:
dy/dt + 150y = 3150
This is a first order linear differential equation in the form dy/dt + p(t)y = g(t), where p(t) = 150 and g(t) = 3150.
The integrating factor e^(∫p(t) dt) = e^(150t).
Multiplying every term of the equation by the integrating factor, we get:
e^(150t) dy/dt + 150e^(150t)y = 3150e^(150t)
The left hand side of this equation is the derivative of (e^(150t)y) with respect to t.
So, we have:
d/dt [e^(150t)y] = 3150e^(150t)
Integrating both sides with respect to t, we get:
e^(150t)y = ∫3150e^(150t) dt + C
Solving the integral on the right hand side, we get:
e^(150t)y = 21e^(150t) + C
Finally, solving for y, we get:
y(t) = 21 + Ce^(-150t)
The constant C can be found using the initial condition y(0) = 95:
95 = 21 + C
So, C = 74.
Therefore, the solution to the differential equation is:
y(t) = 21 + 74e^(-150t)
This is the expression for the temperature of the coffee at time t.
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