**Electric Circuit Analysis Problem** The voltage \( V \) (volts), current \( I \) (amperes), and resistance \( R \) (ohms) of an electric circuit are related by the equation \( V = IR \). Suppose that \( V \) is increasing at the rate of 2 volts/second while \( I \) is decreasing at the rate of \( \frac{1}{4} \) ampere/second. Let \( t \) denote time in seconds. ### Questions: **A. What is the value of \( \frac{dV}{dt} \)?** **B. What is the value of \( \frac{dI}{dt} \)?** **C. Write the equation that relates \( \frac{dV}{dt} \), \( \frac{dI}{dt} \), and \( \frac{dR}{dt} \).** **D. Find the rate of change in \( R \) if \( V = 20 \) volts and \( I = 5 \) amps. Is \( R \) increasing or decreasing?** ### Solution Guide: To answer these questions, follow these steps: **For parts A and B:** - The rate of change of voltage \( \frac{dV}{dt} \) is given as 2 volts/second. - The rate of change of current \( \frac{dI}{dt} \) is given as \( -\frac{1}{4} \) amperes/second (since \( I \) is decreasing). **For part C:** 1. Differentiate the equation \( V = IR \) with respect to time \( t \): \[ \frac{d}{dt}(V) = \frac{d}{dt}(I \cdot R) \] 2. Apply the product rule of differentiation: \[ \frac{dV}{dt} = I \frac{dR}{dt} + R \frac{dI}{dt} \] **For part D:** 1. Substitute the known values into the differentiated equation: \[ 2 = 5 \frac{dR}{dt} + R \left( -\frac{1}{4} \right) \] 2. Solve for \( \frac{dR}{dt} \). **Calculation Steps:** 1.
**Electric Circuit Analysis Problem** The voltage \( V \) (volts), current \( I \) (amperes), and resistance \( R \) (ohms) of an electric circuit are related by the equation \( V = IR \). Suppose that \( V \) is increasing at the rate of 2 volts/second while \( I \) is decreasing at the rate of \( \frac{1}{4} \) ampere/second. Let \( t \) denote time in seconds. ### Questions: **A. What is the value of \( \frac{dV}{dt} \)?** **B. What is the value of \( \frac{dI}{dt} \)?** **C. Write the equation that relates \( \frac{dV}{dt} \), \( \frac{dI}{dt} \), and \( \frac{dR}{dt} \).** **D. Find the rate of change in \( R \) if \( V = 20 \) volts and \( I = 5 \) amps. Is \( R \) increasing or decreasing?** ### Solution Guide: To answer these questions, follow these steps: **For parts A and B:** - The rate of change of voltage \( \frac{dV}{dt} \) is given as 2 volts/second. - The rate of change of current \( \frac{dI}{dt} \) is given as \( -\frac{1}{4} \) amperes/second (since \( I \) is decreasing). **For part C:** 1. Differentiate the equation \( V = IR \) with respect to time \( t \): \[ \frac{d}{dt}(V) = \frac{d}{dt}(I \cdot R) \] 2. Apply the product rule of differentiation: \[ \frac{dV}{dt} = I \frac{dR}{dt} + R \frac{dI}{dt} \] **For part D:** 1. Substitute the known values into the differentiated equation: \[ 2 = 5 \frac{dR}{dt} + R \left( -\frac{1}{4} \right) \] 2. Solve for \( \frac{dR}{dt} \). **Calculation Steps:** 1.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.7: Applications
Problem 16EQ
Related questions
Question
please explain this with solutions!
![**Electric Circuit Analysis Problem**
The voltage \( V \) (volts), current \( I \) (amperes), and resistance \( R \) (ohms) of an electric circuit are related by the equation \( V = IR \). Suppose that \( V \) is increasing at the rate of 2 volts/second while \( I \) is decreasing at the rate of \( \frac{1}{4} \) ampere/second. Let \( t \) denote time in seconds.
### Questions:
**A. What is the value of \( \frac{dV}{dt} \)?**
**B. What is the value of \( \frac{dI}{dt} \)?**
**C. Write the equation that relates \( \frac{dV}{dt} \), \( \frac{dI}{dt} \), and \( \frac{dR}{dt} \).**
**D. Find the rate of change in \( R \) if \( V = 20 \) volts and \( I = 5 \) amps. Is \( R \) increasing or decreasing?**
### Solution Guide:
To answer these questions, follow these steps:
**For parts A and B:**
- The rate of change of voltage \( \frac{dV}{dt} \) is given as 2 volts/second.
- The rate of change of current \( \frac{dI}{dt} \) is given as \( -\frac{1}{4} \) amperes/second (since \( I \) is decreasing).
**For part C:**
1. Differentiate the equation \( V = IR \) with respect to time \( t \):
\[
\frac{d}{dt}(V) = \frac{d}{dt}(I \cdot R)
\]
2. Apply the product rule of differentiation:
\[
\frac{dV}{dt} = I \frac{dR}{dt} + R \frac{dI}{dt}
\]
**For part D:**
1. Substitute the known values into the differentiated equation:
\[
2 = 5 \frac{dR}{dt} + R \left( -\frac{1}{4} \right)
\]
2. Solve for \( \frac{dR}{dt} \).
**Calculation Steps:**
1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa451162e-14ac-4cfd-a2c6-1eae7f4a5f1a%2Fd47ca21b-3b10-487c-af53-01869910b0fa%2Ffg3y5om_processed.png&w=3840&q=75)
Transcribed Image Text:**Electric Circuit Analysis Problem**
The voltage \( V \) (volts), current \( I \) (amperes), and resistance \( R \) (ohms) of an electric circuit are related by the equation \( V = IR \). Suppose that \( V \) is increasing at the rate of 2 volts/second while \( I \) is decreasing at the rate of \( \frac{1}{4} \) ampere/second. Let \( t \) denote time in seconds.
### Questions:
**A. What is the value of \( \frac{dV}{dt} \)?**
**B. What is the value of \( \frac{dI}{dt} \)?**
**C. Write the equation that relates \( \frac{dV}{dt} \), \( \frac{dI}{dt} \), and \( \frac{dR}{dt} \).**
**D. Find the rate of change in \( R \) if \( V = 20 \) volts and \( I = 5 \) amps. Is \( R \) increasing or decreasing?**
### Solution Guide:
To answer these questions, follow these steps:
**For parts A and B:**
- The rate of change of voltage \( \frac{dV}{dt} \) is given as 2 volts/second.
- The rate of change of current \( \frac{dI}{dt} \) is given as \( -\frac{1}{4} \) amperes/second (since \( I \) is decreasing).
**For part C:**
1. Differentiate the equation \( V = IR \) with respect to time \( t \):
\[
\frac{d}{dt}(V) = \frac{d}{dt}(I \cdot R)
\]
2. Apply the product rule of differentiation:
\[
\frac{dV}{dt} = I \frac{dR}{dt} + R \frac{dI}{dt}
\]
**For part D:**
1. Substitute the known values into the differentiated equation:
\[
2 = 5 \frac{dR}{dt} + R \left( -\frac{1}{4} \right)
\]
2. Solve for \( \frac{dR}{dt} \).
**Calculation Steps:**
1.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 6 steps with 10 images
Recommended textbooks for you
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
Glencoe Algebra 1, Student Edition, 9780079039897…
Algebra
ISBN:
9780079039897
Author:
Carter
Publisher:
McGraw Hill
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
Calculus For The Life Sciences
Calculus
ISBN:
9780321964038
Author:
GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:
Pearson Addison Wesley,
Glencoe Algebra 1, Student Edition, 9780079039897…
Algebra
ISBN:
9780079039897
Author:
Carter
Publisher:
McGraw Hill
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage