### Problem Statement Use hyperbolic functions to parametrize the intersection of the surfaces \(x^2 - y^2 = 25\) and \(z = 5xy\). (Use symbolic notation and fractions where needed. Use hyperbolic cosine for parametrization \(x\) variable.) \[ x(t) = 5 \cosh(t) \] \[ y(t) = \sqrt{\cosh^2(t) - 25} \] \[ z(t) = 5 \cosh(t) \sqrt{\cosh^2(t) - 25} \] --- ### Detailed Explanation #### Parametrization using Hyperbolic Functions The given functions make use of hyperbolic cosine (\(\cosh\)) to describe the relationship between \(x\), \(y\), and \(z\) on the intersection of the two surfaces \(x^2 - y^2 = 25\) and \(z = 5xy\). Below, each function is broken down: 1. **Function \(x(t)\)**: \[ x(t) = 5 \cosh(t) \] This equation uses the hyperbolic cosine function to define \(x\) in terms of a parameter \(t\). 2. **Function \(y(t)\)**: \[ y(t) = \sqrt{\cosh^2(t) - 25} \] Here, \(y\) is defined in a way that it satisfies the hyperbolic equation in terms of \(t\). 3. **Function \(z(t)\)**: \[ z(t) = 5 \cosh(t) \sqrt{\cosh^2(t) - 25} \] The equation for \(z\) combines both \(x(t)\) and \(y(t)\) in accordance with the given surface intersection equation \(z = 5xy\). By using the parameter \(t\), it is possible to describe the coordinates \((x, y, z)\) along the intersection of the two surfaces in a smooth and continuous manner.
### Problem Statement Use hyperbolic functions to parametrize the intersection of the surfaces \(x^2 - y^2 = 25\) and \(z = 5xy\). (Use symbolic notation and fractions where needed. Use hyperbolic cosine for parametrization \(x\) variable.) \[ x(t) = 5 \cosh(t) \] \[ y(t) = \sqrt{\cosh^2(t) - 25} \] \[ z(t) = 5 \cosh(t) \sqrt{\cosh^2(t) - 25} \] --- ### Detailed Explanation #### Parametrization using Hyperbolic Functions The given functions make use of hyperbolic cosine (\(\cosh\)) to describe the relationship between \(x\), \(y\), and \(z\) on the intersection of the two surfaces \(x^2 - y^2 = 25\) and \(z = 5xy\). Below, each function is broken down: 1. **Function \(x(t)\)**: \[ x(t) = 5 \cosh(t) \] This equation uses the hyperbolic cosine function to define \(x\) in terms of a parameter \(t\). 2. **Function \(y(t)\)**: \[ y(t) = \sqrt{\cosh^2(t) - 25} \] Here, \(y\) is defined in a way that it satisfies the hyperbolic equation in terms of \(t\). 3. **Function \(z(t)\)**: \[ z(t) = 5 \cosh(t) \sqrt{\cosh^2(t) - 25} \] The equation for \(z\) combines both \(x(t)\) and \(y(t)\) in accordance with the given surface intersection equation \(z = 5xy\). By using the parameter \(t\), it is possible to describe the coordinates \((x, y, z)\) along the intersection of the two surfaces in a smooth and continuous manner.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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I keep getting the wrong answer can you help me
![### Problem Statement
Use hyperbolic functions to parametrize the intersection of the surfaces \(x^2 - y^2 = 25\) and \(z = 5xy\).
(Use symbolic notation and fractions where needed. Use hyperbolic cosine for parametrization \(x\) variable.)
\[
x(t) = 5 \cosh(t)
\]
\[
y(t) = \sqrt{\cosh^2(t) - 25}
\]
\[
z(t) = 5 \cosh(t) \sqrt{\cosh^2(t) - 25}
\]
---
### Detailed Explanation
#### Parametrization using Hyperbolic Functions
The given functions make use of hyperbolic cosine (\(\cosh\)) to describe the relationship between \(x\), \(y\), and \(z\) on the intersection of the two surfaces \(x^2 - y^2 = 25\) and \(z = 5xy\). Below, each function is broken down:
1. **Function \(x(t)\)**:
\[
x(t) = 5 \cosh(t)
\]
This equation uses the hyperbolic cosine function to define \(x\) in terms of a parameter \(t\).
2. **Function \(y(t)\)**:
\[
y(t) = \sqrt{\cosh^2(t) - 25}
\]
Here, \(y\) is defined in a way that it satisfies the hyperbolic equation in terms of \(t\).
3. **Function \(z(t)\)**:
\[
z(t) = 5 \cosh(t) \sqrt{\cosh^2(t) - 25}
\]
The equation for \(z\) combines both \(x(t)\) and \(y(t)\) in accordance with the given surface intersection equation \(z = 5xy\).
By using the parameter \(t\), it is possible to describe the coordinates \((x, y, z)\) along the intersection of the two surfaces in a smooth and continuous manner.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F96584006-fab9-4c6d-81e5-caf655fa5a7f%2Fce6f1fbb-9d44-4bf7-8fd9-2d0689b6e996%2F5jne5uh_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Use hyperbolic functions to parametrize the intersection of the surfaces \(x^2 - y^2 = 25\) and \(z = 5xy\).
(Use symbolic notation and fractions where needed. Use hyperbolic cosine for parametrization \(x\) variable.)
\[
x(t) = 5 \cosh(t)
\]
\[
y(t) = \sqrt{\cosh^2(t) - 25}
\]
\[
z(t) = 5 \cosh(t) \sqrt{\cosh^2(t) - 25}
\]
---
### Detailed Explanation
#### Parametrization using Hyperbolic Functions
The given functions make use of hyperbolic cosine (\(\cosh\)) to describe the relationship between \(x\), \(y\), and \(z\) on the intersection of the two surfaces \(x^2 - y^2 = 25\) and \(z = 5xy\). Below, each function is broken down:
1. **Function \(x(t)\)**:
\[
x(t) = 5 \cosh(t)
\]
This equation uses the hyperbolic cosine function to define \(x\) in terms of a parameter \(t\).
2. **Function \(y(t)\)**:
\[
y(t) = \sqrt{\cosh^2(t) - 25}
\]
Here, \(y\) is defined in a way that it satisfies the hyperbolic equation in terms of \(t\).
3. **Function \(z(t)\)**:
\[
z(t) = 5 \cosh(t) \sqrt{\cosh^2(t) - 25}
\]
The equation for \(z\) combines both \(x(t)\) and \(y(t)\) in accordance with the given surface intersection equation \(z = 5xy\).
By using the parameter \(t\), it is possible to describe the coordinates \((x, y, z)\) along the intersection of the two surfaces in a smooth and continuous manner.
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