Question

Water whose temperature is at 100∘C is left to cool in a room where the temperature is 60∘C. After 3 minutes, the water temperature is 90∘. If the water temperature T is a function of time t given by T = 60 + 40 e k t T=60+40ekt, find the time for the water temperature to reach 65∘C. Round to the nearest hundredth of a minute.

Answer 1

Answer:

21.68 minutes ≈ 21.7 minutes

Step-by-step explanation:

Given:

$T=60+40e^{kt}$

Initial temperature

T = 100°C

Final temperature = 60°C

Temperature after (t = 3 minutes) = 90°C

Now,

using the given equation

$T=60+40e^{kt}$

at T = 90°C and  t = 3 minutes

$90=60+40e^{k(3)}$

$30=40e^{3k}$

or

$e^{3k}=\frac{3}{4}$

taking the natural log both sides, we get

3k = $\ln(\frac{3}{4})$

or

3k = -0.2876

or

k = -0.09589

Therefore,

substituting k in 1 for time at temperature, T = 65°C

$65=60+40e^{( -0.09589)t}$

or

$5=40e^{( -0.09589)t}$

or

$e^{( -0.09589)t}=\frac{5}{40}$

or

$e^{( -0.09589)t}=0.125$

taking the natural log both the sides, we get

( -0.09589)t = ln(0.125)

or

( -0.09589)t = -2.0794

or

t = 21.68 minutes ≈ 21.7 minutes