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![### Question 8
A current of 13 amperes branches into currents \( x \), \( y \), and \( z \) through resistors with resistances 2, 8, and 6 ohms as shown.
#### Diagram Explanation:
- The circuit diagram shows a horizontal line with a current \( I = 13 \) amps entering a junction.
- At the junction, the current splits into three branches:
- The first branch has current \( I = x \) and a resistor with \( R = 2 \) ohms.
- The second branch has current \( I = y \) and a resistor with \( R = 8 \) ohms.
- The third branch has current \( I = z \) and a resistor with \( R = 6 \) ohms.
- These branches rejoin at the other side, completing the circuit.
It is known that the current splits in such a way that the sum of the currents through the three resistors equals the initial current. The energy generated in each resistor is given by
\[ E = I^2R \]
where \( I \) is the current in that resistor and \( R \) is the resistance. Use Lagrange multipliers to find the currents \( x \), \( y \), and \( z \) which will minimize the total energy generated. (It turns out that nature always splits the currents so that the total energy is minimized.)
#### Solution:
Input fields for solutions:
- \( x = \) _______ amps
- \( y = \) _______ amps
- \( z = \) _______ amps
#### Additional Help:
A link or button labeled "Question Help" followed by “[Video]” for additional resources.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9c62a616-6a2f-456f-ac81-c6090d5022b3%2F8606f6c7-5466-4c6b-998e-2371c1deb0b2%2Fs80n1et_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Question 8
A current of 13 amperes branches into currents \( x \), \( y \), and \( z \) through resistors with resistances 2, 8, and 6 ohms as shown.
#### Diagram Explanation:
- The circuit diagram shows a horizontal line with a current \( I = 13 \) amps entering a junction.
- At the junction, the current splits into three branches:
- The first branch has current \( I = x \) and a resistor with \( R = 2 \) ohms.
- The second branch has current \( I = y \) and a resistor with \( R = 8 \) ohms.
- The third branch has current \( I = z \) and a resistor with \( R = 6 \) ohms.
- These branches rejoin at the other side, completing the circuit.
It is known that the current splits in such a way that the sum of the currents through the three resistors equals the initial current. The energy generated in each resistor is given by
\[ E = I^2R \]
where \( I \) is the current in that resistor and \( R \) is the resistance. Use Lagrange multipliers to find the currents \( x \), \( y \), and \( z \) which will minimize the total energy generated. (It turns out that nature always splits the currents so that the total energy is minimized.)
#### Solution:
Input fields for solutions:
- \( x = \) _______ amps
- \( y = \) _______ amps
- \( z = \) _______ amps
#### Additional Help:
A link or button labeled "Question Help" followed by “[Video]” for additional resources.
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