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Answers without the blur. Sign up and see all textbooks for free! Q 3.3-11E

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 # During the summer the temperature inside a van reaches ${\mathbf{55}}{\mathbf{°}}{\mathbf{C}}$, while that outside is a constant ${\mathbf{35}}{\mathbf{°}}{\mathbf{C}}$. When the driver gets into the van, she turns on the air conditioner with the thermostat set at ${\mathbf{16}}{\mathbf{°}}{\mathbf{C}}$. If the time constant for the van is $\frac{\mathbf{1}}{\mathbf{k}}{\mathbf{=}}{\mathbf{2}}{ }{\mathbf{hr}}$ and that for the van with its air conditioning system is $\frac{\mathbf{1}}{{\mathbf{k}}_{\mathbf{1}}}{\mathbf{=}}\frac{\mathbf{1}}{\mathbf{3}}{ }{\mathbf{hr}}$, when will the temperature inside the van reach ${\mathbf{27}}{\mathbf{°}}{\mathbf{C}}$?

The temperature inside the van will reach $27°\mathrm{C}$after $38.04 \mathrm{minutes}$.

See the step by step solution

## Step1: Given data.

The temperature inside a van is $55°\mathrm{C}$and that outside is a constant $35°\mathrm{C}$. When the driver gets into the van, she turns on the air conditioner with the thermostat set at $16°\mathrm{C}$. When the driver gets into the van, she turns on the air conditioner with the thermostat set at $16°\mathrm{C}$. Given the time constant for the van is $\frac{1}{\mathrm{k}}=2 \mathrm{hr}$and that for the van with its air conditioning system is $\frac{1}{{\mathrm{k}}_{1}}=\frac{1}{3} \mathrm{hr}$. It has to find the time after which the temperature inside the van will reach $27°\mathrm{C}$.

## Step 2: Analyzing the given statement

Here, temperature inside the van, ${T}_{in}={55}^{0}C$.

Temperature outside the van, ${T}_{out}={35}^{0}C$.

Temperature value on thermostat, ${T}_{t}={16}^{0}C$.

The time constant for the van is $\frac{1}{\mathrm{k}}=2 \mathrm{hr}$.

The time constant for the van with its air conditioning system is $\frac{1}{{\mathrm{k}}_{1}}=\frac{1}{3} \mathrm{hr}$.

It will use the following formula to find the solution,

$\frac{dT}{dt}={K}_{1}\left({T}_{out}-T\right)+{K}_{u}\left({T}_{t}-T\right)$ …… (1)

## Step 2: To find the value of Ku

As it knows that,

${K}_{1}+{K}_{u}=K$

Using values from step 1,

$3+{K}_{u}=\frac{1}{2}\phantom{\rule{0ex}{0ex}}{K}_{u}=\frac{1}{2}-3\phantom{\rule{0ex}{0ex}}{K}_{u}=\frac{1-6}{2}\phantom{\rule{0ex}{0ex}}{K}_{u}=\frac{-5}{2}\phantom{\rule{0ex}{0ex}}$

It will use this value in equation (1).

## Step 3: To determine the time when the temperature inside the van will reach 27∘C

Now from equation (1),

$\frac{dT}{dt}=3\left(35-T\right)-\frac{5}{2}\left(16-T\right)\phantom{\rule{0ex}{0ex}}\frac{dT}{dt}=\frac{130-T}{2}\phantom{\rule{0ex}{0ex}}\frac{dT}{dt}=65-\frac{T}{2}$

i.e., $\frac{dT}{dt}+\frac{T}{2}=65$ …… (2)

Integrating factor =role="math" localid="1664179724944" ${{\mathbit{e}}}^{\mathbf{\int }\frac{\mathbf{1}}{\mathbf{2}}\mathbf{dt}}{\mathbf{=}}{{\mathbit{e}}}^{\frac{\mathbf{1}}{\mathbf{2}}\mathbf{t}}$

Multiplying both sides of (2) by ${e}^{\frac{1}{2}t}$,

${e}^{\frac{1}{2}t}·\frac{dT}{dt}+{e}^{\frac{1}{2}t}·\frac{T}{2}=65·{e}^{\frac{1}{2}t}\phantom{\rule{0ex}{0ex}}\frac{d}{dt}\left(T·{e}^{\frac{1}{2}t}\right)=65·{e}^{\frac{1}{2}t}\phantom{\rule{0ex}{0ex}}$

Integrating both sides,

$T·{e}^{\frac{1}{2}t}=130{e}^{\frac{1}{2}t}+C$Where, C is an arbitrary constant.

When $\mathrm{t}=0,\mathrm{T}={55}^{\mathrm{o}}\mathrm{C}$

$55=130+C\phantom{\rule{0ex}{0ex}}C=-75$

Therefore,

When temperature is ${{\mathbf{27}}}^{{\mathbf{\circ }}}{\mathbit{C}}$

$27=130-75{e}^{-\frac{1}{2}t}\phantom{\rule{0ex}{0ex}}27-130=-75{e}^{-\frac{1}{2}t}\phantom{\rule{0ex}{0ex}}103=75{e}^{-\frac{1}{2}t}\phantom{\rule{0ex}{0ex}}t=2ln\left(1.373\right)\phantom{\rule{0ex}{0ex}}t=0.634hr\phantom{\rule{0ex}{0ex}}t=38.04min\phantom{\rule{0ex}{0ex}}$

Hence, the temperature inside the van will reach role="math" localid="1664180086516" ${\mathbf{27}}{\mathbf{°}}{\mathbf{C}}$ after role="math" localid="1664180100673" ${\mathbf{38}}{\mathbf{.}}{\mathbf{04}}{ }{ }{\mathbf{minutes}}$. ### Want to see more solutions like these? 