a.
To find: To find the shortest possible horizontal distance from the jet when hearing first sonic boom.
The horizontal distance from the jet is
Given information : The jet makes a sonic boom heard along
Calculation:
When the observer is positioned at the hyperbola's vertex, the horizontal distance between them and the plane is the shortest.
As a result of the plane being at the cone's vertex, the distance between them is equal to the horizontal half-axis of the hyperbola.
Therefore,
Conclusion:
The horizontal distance from the jet when hearing the first sonic boom is
b.
To find: To find the shortest possible horizontal distance from the jet when hearing second sonic boom.
The horizontal distance from the jet is
Given information : The jet makes a sonic boom heard along
Calculation:
Similarly comparing to the part (a), when the observer is positioned at the hyperbola's vertex, the horizontal distance between them and the plane is the shortest.
As a result of the plane being at the cone's vertex, the distance between them is equal to the horizontal half-axis of the hyperbola.
Therefore,
Conclusion:
The horizontal distance from the jet when hearing the first sonic boom is
c.
To find: To describe the relationship between the two hyperbolas.
The first hyperbola is simply the second one scaled in both directions by a factor of
Given information : The two hyperbola’s equations are
Calculation:
The hyperbola equation from the first sonic boom is,
The hyperbola equation from the second sonic boom is,
Comparing the above two hyperbolae’s, to get
As a result, the first hyperbola is half-axes double the length of the second hyperbola.
Conclusion:
The first hyperbola is simply the second one scaled in both directions by a factor of
Chapter 8 Solutions
Holt Mcdougal Larson Algebra 2: Student Edition 2012
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