Question

a thermometer reading 18 F is brought into a room, the temperature of which is 70 F. one minute later the reading is 31 F. determine the temperature reading as a function of time. find the temperatire reading 5 minutes after the thermometer is first brought into the room

Answer 1

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Answer 2

Given:

The temperature of the room (T)= 70°F

The initial temperature of the thermometer (To) = 18°F

The temperature after 1 minute (T₁) = 31°F

To Find:

(i) The temperature reading as a function of time

(ii) The temperature reading after 5 minutes

Solution:

Let θ be the temperature (in ◦F) recorded by the thermometer at time t. The relation between the temperature and time is given by Newton’s Law of Cooling:

dθ / dt = k ( θ - T surroundings)           - (1)

Here k = Newton's constant and T surroundings = 70°F

(i) Integrating equation (1),

θ - 70 = $A e^k^t$

or θ = 70 + $A e^k^t$       - (2)

(ii) Substituting θ = 18°F and t = 0 in equation (2),

18 = 70 + Ae⁰

or A = -52

Hence, θ = 70 - 52 $e^K^t$        - (3)

Now substituting θ = 31°F and t = 1 in equation (3),

31 = 70 - 52 $e^k$

or $e^k = \frac{39}{52}$

Taking ln o both sides,

k = ln (39/52)

Thus θ = 70 - 52 X $(39/52) ^t$        - (4)

Substituting, t = 5 in eqution 4,

θ = 70 - 52 X 0.75⁵

or θ = 57.67 °F

Hence, the relation between the temperature reading and time is $\theta = 70 - 52 e^k^t$ and the temperature after 5 minutes is 57.67 °F