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Chapter 6: Problem 3

Mileage for an old car: The gas mileage \(M\) that you get on your car depends on its age \(t\) in years. a. Explain the meaning of \(\frac{d M}{d t}\) in practical terms. b. As your car ages and performance degrades, do you expect \(\frac{d M}{d t}\) to be positive or negative?

Short Answer

Expert verified
\(\frac{d M}{d t}\) represents how mileage changes with age. It is negative as mileage decreases with age.

Step by step solution

01

Understanding the Derivative

The expression \(\frac{d M}{d t}\) represents the derivative of the gas mileage \(M\) with respect to the car's age \(t\). In practical terms, this means it is the rate at which the gas mileage changes as the car gets older. It tells us how much the mileage is increasing or decreasing for each passing year.
02

Interpreting Performance Degradation

As cars age, various factors such as engine wear, decreased efficiency, and general degradation tend to decrease overall performance. This usually results in decreased gas mileage. Therefore, we expect \(\frac{d M}{d t}\) to be negative since mileage decreases as the car gets older.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Gas Mileage
Gas mileage is a critical measure of a car's fuel efficiency. It indicates how many miles a vehicle can travel per gallon of fuel. Generally, a higher gas mileage is desirable as it means the car consumes less fuel over a given distance. This is good news for both your wallet and the environment. For older cars, understanding gas mileage becomes even more important because it typically decreases as the car ages.

There are many factors influencing the gas mileage of an old car:
  • Engine Condition: Worn-out engines may burn fuel inefficiently, reducing miles per gallon.
  • Maintenance: Irregular upkeep can lead to poor vehicle performance impacting efficiency.
  • Driving Habits: Aggressive driving or frequent short trips can negatively affect mileage over time.
By measuring and understanding gas mileage, you can take steps to improve car efficiency and make informed decisions about maintenance and potential replacements.
Rate of Change and Its Importance
The rate of change in calculus, often described as a derivative, helps measure how one quantity changes in relation to another. In the case of gas mileage, \(\frac{d M}{d t}\) represents the rate of change of mileage with respect to time, specifically the car's age in years.

Understanding this rate of change is crucial for:
  • Forecasting: Predict future performance issues, helping plan maintenance or upgrades.
  • Efficiency Monitoring: Keep track of when and how performance slips, identifying possible mechanical problems early.
  • Making Decisions: Decide when it might be time to retire an aging vehicle based on declining efficiency.
The principle of rate of change provides valuable insights into the continuous function of gas mileage over time, allowing for better management of vehicle performance.
Impact of Car Performance Degradation
As a car ages, its overall performance often deteriorates. This degradation can be attributed to several factors including natural wear and tear, component failures, and outdated technology. Car performance degradation typically leads to a decline in gas mileage, indicating increased fuel consumption for the same travel distance.

Some key points regarding performance degradation include:
  • Component Wear: Parts like pistons, rings, and valves wear out, leading to engine inefficiencies.
  • Age-related Issues: Older cars may not have advanced fuel efficiency technologies of newer models.
  • Repair Costs: As performance declines, repair costs may increase, counterbalancing savings from higher fuel consumption.
By being aware of how car performance naturally degrades over time, you can prioritize important repairs and maintenance efforts, maintaining better control over a vehicle's operational costs and efficiency.

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Most popular questions from this chapter

The cannon with a different muzzle velocity: If the cannonball from Example 6.7 is fired with a muzzle velocity of 370 feet per second, it will follow the graph of $$ h=x-32\left(\frac{x}{370}\right)^{2} $$ where distances are measured in feet. a. Plot the graph of the flight of the cannonball. b. Find the height of the cannonball 3000 feet downrange. c. By looking at the graph of \(h\), determine whether \(\frac{d h}{d x}\) is positive or negative at 3000 feet downrange. d. Calculate \(\frac{d h}{d x}\) at 3000 feet downrange and explain what this number means in practical terms.

Sprinkler irrigation in Nebraska: Logistic growth can be used to model not only population growth but also economic and other types of growth. For example, the total number of acres \(A=A(t)\), in millions, in Nebraska that are being irrigated by modern sprinkler systems has shown approximate logistic growth since 1955 , closely following the equation of change $$ \frac{d A}{d t}=0.15 A\left(1-\frac{A}{3}\right) \text {. } $$ Here time \(t\) is measured in years. a. According to this model, how many total acres in Nebraska can be expected eventually to be irrigated by sprinkler systems? (Hint: This corresponds to the carrying capacity in the logistic model for population growth.) b. How many acres of land were under sprinkler irrigation when sprinkler irrigation was expanding at its most rapid rate?

Radioactive decay: The amount remaining \(A\), in grams, of a radioactive substance is a function of time \(t\), measured in days since the experiment began. The equation of change for \(A\) is $$ \frac{d A}{d t}=-0.05 A $$ a. What is the exponential growth rate for \(A\) ? b. If initially there are 3 grams of the substance, find a formula for \(A\). c. What is the half-life of this radioactive substance?

Traveling in a car: Make graphs of location and velocity for each of the following driving events. In each case, assume that the car leaves from home moving west down a straight road and that position is given as the distance west from home. a. A vacation: Being eager to begin your overdue vacation, you set your cruise control and drive faster than you should to the airport. You park your car there and get on an airplane to Spain. When you fly back 2 weeks later, you are tired and drive at a leisurely pace back home. (Note: Here we are talking about location of your car, not of the airplane.) b. On a country road: A car driving down a country road encounters a deer. The driver slams on the brakes and the deer runs away. The journey is cautiously resumed. c. At the movies: In a movie chase scene, our hero is driving his car rapidly toward the bad guys. When the danger is spotted, he does a Hollywood 180-degree turn and speeds off in the opposite direction.

Making up a story about a car trip: You begin from home on a car trip. Initially your velocity is a small positive number. Shortly after you leave, velocity decreases momentarily to zero. Then it increases rapidly to a large positive number and remains constant for this part of the trip. After a time, velocity decreases to zero and then changes to a large negative number. a. Make a graph of velocity for this trip. b. Discuss your distance from home during this driving event, and make a graph. c. Make up a driving story that matches this description.
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