Book edition 9th

Author(s) R. Kent Nagle, Edward B. Saff, Arthur David Snider

Pages 616 pages

ISBN 9780321977069

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Short Answer

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

The temperature inside the van will reach $27 ° C$ after $38.04 minutes$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step1: Given data.

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

Step 2: Analyzing the given statement

Here, temperature inside the van, $T_{i n} = 55^{0} C$ .

Temperature outside the van, $T_{o u t} = 35^{0} C$ .

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

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

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

It will use the following formula to find the solution,

$\frac{d T}{d t} = K_{1} (T_{o u t} - T) + K_{u} (T_{t} - T)$ …… (1)

Step 2: To find the value of Ku

As it knows that,

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

Using values from step 1,

$3 + K_{u} = \frac{1}{2} K_{u} = \frac{1}{2} - 3 K_{u} = \frac{1 - 6}{2} K_{u} = \frac{- 5}{2}$

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),

$\frac{d T}{d t} = 3 (35 - T) - \frac{5}{2} (16 - T) \frac{d T}{d t} = \frac{130 - T}{2} \frac{d T}{d t} = 65 - \frac{T}{2}$

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

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

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

$e^{\frac{1}{2} t} \cdot \frac{d T}{d t} + e^{\frac{1}{2} t} \cdot \frac{T}{2} = 65 \cdot e^{\frac{1}{2} t} \frac{d}{d t} (T \cdot e^{\frac{1}{2} t}) = 65 \cdot e^{\frac{1}{2} t}$

Integrating both sides,

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

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

$55 = 130 + C C = - 75$

Therefore,

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

$27 = 130 - 75 e^{- \frac{1}{2} t} 27 - 130 = - 75 e^{- \frac{1}{2} t} 103 = 75 e^{- \frac{1}{2} t} t = 2 l n (1.373) t = 0.634 h r t = 38.04 m i n$

Hence, the temperature inside the van will reach role="math" localid="1664180086516" $27 ° C$ after role="math" localid="1664180100673" $38.04 minutes$ .

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Fundamentals Of Differential Equations And Boundary Value Problems

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Step by Step Solution

Step1: Given data.

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Fundamentals Of Differential Equations And Boundary Value Problems

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