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Question 1185983: When a fighter jet travels faster than sound, it generates a sonic boom shock wave in the shape of a cone. If the plane is flying at a constant altitude, this cone intersects the ground in the shape of one branch of a hyperbola. Suppose a jet is flying north at level altitude and at a speed of 500 m/s (which is super sonic speed). In this model, let the position of the jet be the origin. The vertex of the sonic boom hyperbola is on the ground 15km behind the jet, and the hyperbola has an eccentricity of 1.5. Use a scale of 1 unit = 1 km
A.) Find the coordinates of the focus and the equations of the asymptotes. Round all answers to the nearest hundredth.
B.) Write an equation for the sonic boom hyperbola in standard form.
C.) A person on the ground will hear the sonic boom when the hyperbola passes over him. Suppose the jet is located at the origin at time t=0. The jet and the sonic boom line are moving due north at 500 m/s, and John is standing at (7, 10). Calculate the time, in seconds, until John will hear the sonic boom. Round your answer to the nearest tenth of a second
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to solve this problem:
**A. Find the coordinates of the focus and the equations of the asymptotes:**
1. **Hyperbola Orientation:** Since the jet is flying north and the vertex is behind it, the hyperbola opens north (upwards in our coordinate system). This means the hyperbola's transverse axis is vertical.
2. **Center:** The center of the hyperbola is halfway between the jet (origin) and the vertex. Since the vertex is 15 km behind (south of) the jet, the center is 15/2 = 7.5 km south of the jet. Therefore, the center is at (0, -7.5).
3. **Distance from Center to Vertex (a):** The distance from the center to the vertex is 7.5 km. So, a = 7.5.
4. **Distance from Center to Focus (c):** We're given that the eccentricity (e) is 1.5. Eccentricity is defined as e = c/a. Therefore:
1.5 = c / 7.5
c = 1.5 * 7.5 = 11.25
5. **Focus:** The focus is 'c' units from the center along the transverse axis. Since the hyperbola opens upwards, the focus is 11.25 km south of the jet. Therefore, the focus is at (0, -11.25).
6. **Equations of the Asymptotes:** For a hyperbola with a vertical transverse axis, the equations of the asymptotes are:
y - k = ±(a/b)(x - h)
where (h, k) is the center, 'a' is the distance from the center to the vertex, and 'b' is related to 'a' and 'c' by the equation c² = a² + b².
First, we find b:
11.25² = 7.5² + b²
b² = 126.5625 - 56.25
b² = 70.3125
b = √70.3125 ≈ 8.385
Now, substitute the values into the asymptote equations:
y + 7.5 = ±(7.5/8.385)(x - 0)
y ≈ ±0.894x - 7.5
**B. Equation of the Hyperbola:**
The standard form equation for a hyperbola with a vertical transverse axis and center (h, k) is:
((y - k)² / a²) - ((x - h)² / b²) = 1
Substituting the values we found:
((y + 7.5)² / 7.5²) - (x² / 70.3125) = 1
((y + 7.5)² / 56.25) - (x² / 70.3125) = 1
**C. Time until John hears the sonic boom:**
1. **Equation of the Hyperbola's Leading Edge:** The leading edge of the sonic boom is represented by the *right* branch of the hyperbola. Since John is at (7,10), we can plug in x=7 and solve for the y coordinate of the hyperbola's leading edge.
((10 + 7.5)² / 56.25) - (7² / 70.3125) = 1
(17.5²/56.25) - (49/70.3125) = 1
5.44 - 0.697 = 4.743
y = 10.743
2. **Distance Traveled by the Sonic Boom:** The leading edge of the hyperbola is at approximately y = 10.743 km when x=7. The hyperbola is moving north at 500m/s. The distance that the sonic boom has traveled is approximately 10.743 km + 7.5 km = 18.243km.
3. **Time:** Time = Distance / Speed = 18243 m / 500 m/s ≈ 36.5 seconds
**Answers:**
* **A.** Focus: (0, -11.25). Asymptotes: y ≈ ±0.894x - 7.5
* **B.** ((y + 7.5)² / 56.25) - (x² / 70.3125) = 1
* **C.** Time ≈ 36.5 seconds
Answer by ikleyn(52790)  (Show Source):
You can put this solution on YOUR website! .
The "solution" by @CPhill is full of wrong statements and is TOTALLY INCORRECT.
Below I placed his solution (a copy), and marked the places where his statements are incorrect
(practically, everywhere).
************************************************************************
below is my copy-past of the solution by @CPhill
with my comments/marks where the statements
are incorrect.
*********************** S T A R T **************************************
Here's how to solve this problem:
**A. Find the coordinates of the focus and the equations of the asymptotes:**
1. **Hyperbola Orientation:** Since the jet is flying north and the vertex is behind it, the hyperbola opens north (upwards in our coordinate system). This means the hyperbola's transverse axis is vertical.
2. **Center:** The center of the hyperbola is halfway between the jet (origin) and the vertex. Since the vertex is 15 km behind (south of) the jet, the center is 15/2 = 7.5 km south of the jet. Therefore, the center is at (0, -7.5).
^--------------------^
WRONG STATEMENT
+--------------------+
3. **Distance from Center to Vertex (a):** The distance from the center to the vertex is 7.5 km. So, a = 7.5.
^--------------------^
WRONG STATEMENT
+--------------------+
4. **Distance from Center to Focus (c):** We're given that the eccentricity (e) is 1.5. Eccentricity is defined as e = c/a. Therefore:
1.5 = c / 7.5
c = 1.5 * 7.5 = 11.25
^--------------------^
WRONG STATEMENT
+--------------------+
5. **Focus:** The focus is 'c' units from the center along the transverse axis. Since the hyperbola opens upwards, the focus is 11.25 km south of the jet. Therefore, the focus is at (0, -11.25).
^--------------------^
WRONG STATEMENT
+--------------------+
6. **Equations of the Asymptotes:** For a hyperbola with a vertical transverse axis, the equations of the asymptotes are:
y - k = ±(a/b)(x - h)
where (h, k) is the center, 'a' is the distance from the center to the vertex, and 'b' is related to 'a' and 'c' by the equation c² = a² + b².
First, we find b:
11.25² = 7.5² + b²
b² = 126.5625 - 56.25
b² = 70.3125
b = √70.3125 ≈ 8.385
Now, substitute the values into the asymptote equations:
y + 7.5 = ±(7.5/8.385)(x - 0)
y ≈ ±0.894x - 7.5
^--------------------^
WRONG STATEMENT
+--------------------+
**B. Equation of the Hyperbola:**
The standard form equation for a hyperbola with a vertical transverse axis and center (h, k) is:
((y - k)² / a²) - ((x - h)² / b²) = 1
Substituting the values we found:
((y + 7.5)² / 7.5²) - (x² / 70.3125) = 1
((y + 7.5)² / 56.25) - (x² / 70.3125) = 1
^--------------------^
WRONG STATEMENT
+--------------------+
**C. Time until John hears the sonic boom:**
1. **Equation of the Hyperbola's Leading Edge:** The leading edge of the sonic boom is represented by the *right* branch of the hyperbola. Since John is at (7,10), we can plug in x=7 and solve for the y coordinate of the hyperbola's leading edge.
((10 + 7.5)² / 56.25) - (7² / 70.3125) = 1
(17.5²/56.25) - (49/70.3125) = 1
5.44 - 0.697 = 4.743
y = 10.743
2. **Distance Traveled by the Sonic Boom:** The leading edge of the hyperbola is at approximately y = 10.743 km when x=7. The hyperbola is moving north at 500m/s. The distance that the sonic boom has traveled is approximately 10.743 km + 7.5 km = 18.243km.
^----------------------------------------^
WRONG and NON-SENSICAL CALCULATIONS
+----------------------------------------+
3. **Time:** Time = Distance / Speed = 18243 m / 500 m/s ≈ 36.5 seconds
^--------------------^
WRONG STATEMENT
+--------------------+
**Answers:**
* **A.** Focus: (0, -11.25). Asymptotes: y ≈ ±0.894x - 7.5
* **B.** ((y + 7.5)² / 56.25) - (x² / 70.3125) = 1
* **C.** Time ≈ 36.5 seconds
^-------------------------^
EVERYTHING is WRONG
+-------------------------+
*********************** E N D **************************************
//////////////////////////////
The " solution " by @CPhill is full of wrong statements and is TOTALLY INCORRECT.
The solution by @CPhill is a " copy-paste " of the solution by the artificial intelligence, which was incorrect.
The fact that @PChill' solutions are copy-paste from AI is not a discovery to me.
The fact that AI often produces incorrect solutions also is not a discovery to me.
The discovery is that this person, @CPhill, even does not read his " solutions " produced by AI,
does not care about their correctness and DOES NOT understand these solutions COMPLETELY, at all.
May be, later I will place a correct solution here.
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To the tutor @CPhill.
Dear tutor @CPhill,
the way how you present your "solutions" is not quite honest.
To be fully honest in front of the reader, you should write this acknowledgment at the end of each your post:
The solution in this my post is not really mine: it is produced by the current Google AI.
The current Google AI is experimental and may have errors.
I am not responsible for all possible errors - - - the Google AI is responsible for them.
Thank you for your hard work on checking possible errors that will be accounted by the Google AI.
Then everything will be clear, everything will be honest and everything will be correct.
Otherwise, your posts will be obscenity.
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