• :00Days
• :00Hours
• :00Mins
• 00Seconds
A new era for learning is coming soon Suggested languages for you:

Europe

Answers without the blur. Sign up and see all textbooks for free! Q 3.3-7E

Expert-verified Found in: Page 108 ### Fundamentals Of Differential Equations And Boundary Value Problems

Book edition 9th
Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider
Pages 616 pages
ISBN 9780321977069 # On a hot Saturday morning while people are working inside, the air conditioner keeps the temperature inside the building at ${\mathbf{24}}{\mathbf{°}}{\mathbf{C}}$. At noon the air conditioner is turned off, and the people go home. The temperature outside is a constant ${\mathbf{35}}{\mathbf{°}}{\mathbf{C}}$ for the rest of the afternoon. If the time constant for the building is 4 hr, what will be the temperature inside the building at 2:00 p.m.? At 6:00 p.m.? When will the temperature inside the building reach ${\mathbf{27}}{\mathbf{°}}{\mathbf{C}}$?

The temperature inside the building will be $28.3°\mathrm{C}$ at 2:00 p.m. and $32.5°\mathrm{C}$ at 6:00 p.m. The temperature inside the building will reach $27°\mathrm{C}$ after 1.16 pm.

See the step by step solution

## Step1: important formula.

Newton’s Law of cooling is, ${\mathbit{T}}\left(t\right){\mathbf{=}}{\mathbit{M}}{\mathbf{+}}{\mathbit{C}}{{\mathbit{e}}}^{\mathbf{-}\mathbf{k}\mathbf{t}}$

## Step 2: Analyzing the given statement.

The temperature inside the building is $24°\mathrm{C}$. The temperature outside is a constant 35°C for the rest of the afternoon. If the time constant for the building is 4 hr. it has to find the temperature inside the building at 2:00 p.m. and at 6:00 p.m. Also, we have to find the time when the temperature will reach $27°\mathrm{C}$.

Newton’s Law of cooling is,

$T\left(t\right)=M+C{e}^{-kt}$ …… (1)

Here, it will take the values as,

Initial temperature,${T}_{0}={24}^{o}C$,

Constant temperature outside the room, $M={35}^{o}C$.

Time constant for the building is 4 hr i.e., $\frac{1}{k}=4$.

## Step 2: To find the value of C in the formula of Newton’s Law of cooling to find the temperature inside the building at time, t

Using the given values in equation (1), to find the value of ,

So, at t=0,

$T\left(0\right)=35+C{e}^{0}\phantom{\rule{0ex}{0ex}}24=35+C\phantom{\rule{0ex}{0ex}}C=-11\phantom{\rule{0ex}{0ex}}$

Thus, the temperature inside the building at time, t is

$T\left(t\right)=M-11{e}^{-\frac{t}{4}}$ .....................(2)

## Step 3: To find the temperature inside the building at 2:00 p.m.

Substitute t=2 and $\mathrm{M}=35°\mathrm{C}$ in equation (2),

$T\left(2\right)=35-11{e}^{\mathbf{-}\frac{\mathbf{2}}{\mathbf{4}}}\phantom{\rule{0ex}{0ex}}T\left(2\right)=28.{3}^{o}C\phantom{\rule{0ex}{0ex}}$

Hence, the temperature inside the building at 2:00 p.m. will be 28.3°C.

## Step 4: To find the temperature inside the building at 6:00 p.m.

Substitute $\mathrm{t}=6\mathrm{and}\mathrm{M}={35}^{\mathrm{o}}$ in equation (2),

$\mathrm{T}\left(6\right)=35-11{\mathrm{e}}^{-\frac{6}{4}}\phantom{\rule{0ex}{0ex}}\mathrm{T}\left(2\right)=32.{5}^{\mathrm{o}}\mathrm{C}\phantom{\rule{0ex}{0ex}}$

So, the temperature inside the building at 6:00 p.m. will be 32.5°C.

## Step 5: To find the time at which the temperature inside the building will reach 27°C

Substitute $\mathrm{T} \left(\mathrm{t}\right)=2{7}^{\mathrm{o}}\mathrm{C} \mathrm{and} \mathrm{M}={35}^{\mathrm{o}}\mathrm{C}$in equation (2),

$27=35-11{\mathrm{e}}^{-\frac{\mathrm{t}}{4}}\phantom{\rule{0ex}{0ex}}8=\left(11\right){\mathrm{e}}^{-\frac{\mathrm{t}}{4}}\phantom{\rule{0ex}{0ex}}{\mathrm{e}}^{-\frac{\mathrm{t}}{4}}=\frac{8}{11}\phantom{\rule{0ex}{0ex}}-\frac{\mathrm{t}}{4}=\mathrm{ln}\left(0.727\right)\phantom{\rule{0ex}{0ex}}$

Therefore, the temperature inside the building will reach 27°C after 1.16 p.m. ### Want to see more solutions like these? 