Question

Anne Kates house is 20c. She had just made a fresh cup of tea (100c). Five minutes after she made her mad scientist nephew came in,stuck a thermometer in the cup and announced that the tea was now only 70c Write the possible equation with asymptotes

Answer 1

The possible equation is; T(t) = 20 + 80·
`e^((-0.094 \cdot t))`

The horizontal asymptotes is Lim(t → ∞), T → 20°C

The vertical asymptote is Lim(t → 0) T → 100°C

Explanation:

The given parameters are;

The temperature of the house = 20°C

The initial temperature of the tea = 100°C

The temperature of the tea after 5 minutes = 70°C

From Newton's law of cooling, is given as follows;

`T(t) = T_s + (T_0 - T_s) \cdot e^(k \cdot t)`

`(dT)/(dt) = -k(T - T_a)`

Where;

T(t) = The temperature after time, t =

`T_s` =The room temperature = 20°C

T₀ = The initial temperature of the tea = 100°C

k = Constant

T(5) = 70°C, therefore, we have;

`70 = 20 + (100 - 20) \cdot e^(k \cdot 5)`

50/80 =
`e^(k \cdot 5)`

5·k = ㏑(5/8)

k = ㏑(5/8)/5 ≈ -0.094

Therefore, the possible equation is given as follows;

T(t) = 20 + 80·
`e^((-0.094 \cdot t))`

The asymptotes are Lim(t → ∞), T → 20°C and Lim(t → 0) T → 100°C