PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS)
PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS)
8th Edition
ISBN: 9781337904285
Author: Larson
Publisher: CENGAGE L
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Chapter 5.5, Problem 124E

(a)

To determine

To calculate: The angle θ that corresponds to a given Mach number.

(a)

Expert Solution
Check Mark

Answer to Problem 124E

The angle θ that corresponds to a given Mach number is 180° .

Explanation of Solution

Given information:

The following relation:

  sinθ2=1M

Where M correspond to Mach number and M=1

Calculation:

The angle θ that corresponds to the given Mach number is as follows:

  sinθ2=1M         [Given]sinθ2=11   [Substitute]sinθ2=1   [Simplify]θ2=arcsin(1)     [Inverse trigonometric function]θ2=90°  [Value of inverse]θ=180°  [Multiply both sides by 2]

(b)

To determine

To calculate: The angle θ that corresponds to a given Mach number.

(b)

Expert Solution
Check Mark

Answer to Problem 124E

The angle θ that corresponds to a given Mach number is 25.68° .

Explanation of Solution

Given information:

The following relation:

  sinθ2=1M

Where M correspond to Mach number and M=4.5

Calculation:

The angle θ that corresponds to the given Mach number is as follows:

  sinθ2=1M              [Given]sinθ2=14.5        [Substitute]θ2=arcsin(14.5)     [Inverse trigonometric function]θ2=12.84°         [Value of inverse]θ=25.68°         [Multiply both sides by 2]

(c)

To determine

To calculate: The speed of an object having the Mach numbers in part(a) and (b).

(c)

Expert Solution
Check Mark

Answer to Problem 124E

For M=1 , speed of object is 760 miles per hour .

For M=4.5 , speed of object is 3240 miles per hour .

Explanation of Solution

Given information:

The Mach number M of an airplane is the ratio of its speed to the speed of sound.

Speed of sound in air = 760 miles per hour .

The following relation:

  sinθ2=1M

Where M correspond to Mach number.

Calculation:

As given,

  M=VoVs

Where Vo is the speed of object and Vs is the speed of sound.

For M=1 , speed of object is calculated as:

  M=V0Vs                            [Given]1=V0760                             [Substitute]V0=760 miles per hour    [Simplify] 

For M=4.5 , speed of object is calculated as:

  M=V0Vs                               [Given]4.5=V0760                            [Substitute]V0=3420 miles per hour     [Simplify] 

(d)

To determine

To write: The given equation as a trigonometric function of θ .

(d)

Expert Solution
Check Mark

Answer to Problem 124E

The given equation as a trigonometric function of θ can be written as cosθ=12M2 .

Explanation of Solution

Given information:

The following relation:

  sinθ2=1M

Where M correspond to Mach number.

Formula used:

The following half angle formula:

  sinu2=±1cosu2

Calculation:

The given trigonometric equation as a function of θ can be written as:

  sinθ2=1M                [Given]±1cosθ2=1M      [Half angle formula]1cosθ2= 1M2       [Square both sides]1cosθ=2M2         [Multiply both sides by 2]cosθ=12M2         [Simplify]

Chapter 5 Solutions

PRECALCULUS W/LIMITS:GRAPH.APPROACH(HS)

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