12.7. Practice Problems#

12.7.1. Learning Objectives#

After studying this notebook, completing the activities, engaging in class, and reading the book, you should be able to:

  • Apply error propagation strategies to example problems

import numpy as np
import matplotlib.pyplot as plt

12.7.2. Practice Problem 1: One Uncertain Parameter#

Reference: §3.3, Problem 8 in Navidi.

The refractive index \(n\) of a piece of glass is related to the critical angle \(\theta\) by \(n=1/\sin \theta\) Assume that the critical angle is measured to be 0.70 \(\pm\) 0.02 rad. Estimate the refractive index, and find the uncertainty in the estimate.

# Add your solution here

12.7.3. Practice Problem 2: Three Uncertain Parameters#

Reference: §3.4, Problem 8 in Navidi.

The pressure \(P\), temperature \(T\), and volume \(V\) of one mole of an ideal gas are related by the equation \(PV=RT\) where \(P\) is measured in kilopascals, \(T\) is measured in kelvins, and \(V\) is measured in liters, and \(R = 8.314\) with consistent units.

  1. Assume that \(P\) = 242.52 \(\pm\) 0.03 kPa and \(V\) = 10.103 \(\pm\) 0.002 L. Estimate \(T\), and find the uncertainty in the estimate.

  2. Assume that \(P\) = 242.52 \(\pm\) 0.03 kPa and \(T\) = 290.11 \(\pm\) 0.02 K. Estimate \(V\), and find the uncertainty in the estimate.

  3. Assume that \(V\) = 10.103 \(\pm\) 0.002 L and \(T\) = 290.11 \(\pm\) 0.02 K. Estimate \(P\), and find the uncertainty in the estimate.

  4. Repeat calculation 1 assuming \(P\) and \(V\) are not independent, but have a covariance of 10\(^{-5}\) kPa-L.

# Part 1
# Add your solution here

# Part 2
# Add your solution here

# Part 3
# Add your solution here

# Part 4
# Add your solution here

12.7.4. Practice Problem 3: Uncertainty Simulation#

Reference: §3.4, Problem 16 in Navidi.

According to Newton’s law of cooling, the time \(t\) needed for an object at an initial temperature \(T_0\) to cool to a temperature \(T\) in an environment with ambient temperature \(T_a\) is given by

\[t = \frac{\ln(T_0 - T_a)}{k} - \frac{\ln(T - T_a)}{k}\]

where \(k\) is a constant. Assume that for a certain type of container, \(k\) = 0.025 min\(^{-1}\). Let \(t\) be the number of minutes needed to cool the container to a temperature of 50\(^\circ{}\)F. Assume \(T_0\) = 70.1 \(\pm\) 0.2\(^\circ{}\)F and \(T_a\) = 35.7 \(\pm\) 0.1\(^\circ{}\)F. Estimate \(t\), and find the uncertainty in the estimate.

# Define function and calculate t

# Add your solution here

# Estimate gradient with finite difference
# Add your solution here

# Assemble covariance matrix and apply error propagation formula

# Add your solution here

12.7.5. Practice Problem 4: Probability Simulation#

Reference: §4.12, Problem 18 in Navidi.

The length of time to perform an oil change at a certain shop is normally distributed with mean 29.5 minutes and standard deviation 3 minutes. What is the probability that a mechanic can complete 16 oil changes in an eight-hour day?

# Let's use simulation! We'll learn how to calculate this answer by hand next class.

# Add your solution here