An object with initial temperature 130 ∘ f is submerged in large tank of water whose temperature is 50 ∘ f . find a formula for f ( t ) , the temperature of the object after t minutes, if the cooling constant is k = − 0.2 . remember newton's law of cooling (the rate of change of temperature with respect to time is equal to k times the difference between the temperature of the object and the surrounding temperature) ! : )

The formula for the object cooling is , where is in minutes.

The object is cooled by heat mechanism of Convection, Convection is a Heat Transfer mechanism in which is a solid object is cooled due to a fluid in motion and is described by the Newton's Law of Cooling, whose Differential Equation is:

(1)

Where:

- Temperature of the solid, in degrees Fahrenheit.

- Cooling rate, in .

- Water temperature, in degrees Fahrenheit.

The solution of this Differential Equation is:

(2)

Where is the initial temperature of the solid, in degrees Fahrenheit.

If we know that , and , then the formula for the object cooling is:

The formula for the object cooling is , where is in minutes.

Here is a question related to the Newton's Law of Cooling: link

T = 50 + 80e^(-0.2t)

Explanation:

Newton's law of cooling says the rate of change of temperature with respect to time is proportional to the temperature difference:

dT/dt = k (T − Tₐ)

Separating the variables and integrating:

dT / (T − Tₐ) = k dt

ln (T − Tₐ) = kt + C

T − Tₐ = Ce^(kt)

T = Tₐ + Ce^(kt)

Given that Tₐ = 50 and k = -0.2:

T = 50 + Ce^(-0.2t)

At t = 0, T = 130.

130 = 50 + Ce^(0)

130 = 50 + C

C = 80

Therefore:

T = 50 + 80e^(-0.2t)

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