Joan Thompson

2021-12-27

A thermometer reading ${75}^{\circ}F$ is taken out where the temperature is ${20}^{\circ}F$ . The reading is ${30}^{\circ}F$ 4 min later.

A. What is the value of k in four decimal places?

B. Find the thermometer reading 7 min after the thermometer was brought outside.

C. Find the time taken for the reading to drop from${75}^{\circ}F$ to within a half degree of the air temperature.

A. What is the value of k in four decimal places?

B. Find the thermometer reading 7 min after the thermometer was brought outside.

C. Find the time taken for the reading to drop from

boronganfh

Beginner2021-12-28Added 33 answers

Step 1

Given that:

- Initial temperature$={75}^{\circ}F$

- Surrounding temperature$={20}^{\circ}F$

- The temperature is${30}^{\circ}F$ after 4 minute

Step 2

According to the Newtons

Given that:

- Initial temperature

- Surrounding temperature

- The temperature is

Step 2

According to the Newtons

ambarakaq8

Beginner2021-12-29Added 31 answers

Solution:

$\frac{dT}{dt}=-k(T-{T}_{s})$

$\frac{dT}{dt}=-k(T-20)$

$\frac{dT}{T-20}=-k\text{}dt$

$\mathrm{ln}(T-20)=-kt+\mathrm{ln}C$

$\mathrm{ln}(T-20)={\mathrm{ln}e}^{-kt}+\mathrm{ln}C$

$\mathrm{ln}(T-20)=\mathrm{ln}C{e}^{-kt}$

$T-20=C{e}^{-kt}$

$T=20+C{e}^{-kt}$

When$t=0,T={75}^{\circ}$

$75=20+C{e}^{0}$

$C=55$

When$t=4,T={30}^{\circ}$

$30=20+55{e}^{-4k}$

$\frac{10}{55}={e}^{-4k}$

$e}^{-k}={\left(\frac{2}{11}\right)}^{\frac{1}{4}$

Thus,$T=20+55{\left(\frac{2}{11}\right)}^{\frac{t}{4}}$

When$T={20.5}^{\circ}F$

$20.5=20+55{\left(\frac{2}{11}\right)}^{\frac{t}{4}}$

$\frac{0.5}{55}={\left(\frac{2}{11}\right)}^{\frac{t}{4}}$

$\mathrm{ln}\frac{0.5}{55}={\mathrm{ln}\left(\frac{2}{11}\right)}^{\frac{t}{4}}$

$\mathrm{ln}\left(\frac{1}{110}\right)=\frac{1}{4}t\mathrm{ln}\left(\frac{2}{11}\right)$

$t=\frac{4\mathrm{ln}\left(\frac{1}{110}\right)}{\mathrm{ln}\left(\frac{2}{11}\right)}$

$t=11.029\text{}minutes$

When

When

Thus,

When

Vasquez

Expert2022-01-09Added 669 answers

user_27qwe

Skilled2023-05-14Added 375 answers

karton

Expert2023-05-14Added 613 answers

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