Textbook Question
Find the height h, radius r, and volume of a right circular cylinder with maximum volume that is inscribed in a sphere of radius R.
Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 4, Problem 49e
Travel costs A simple model for travel costs involves the cost of gasoline and the cost of a driver. Specifically, assume gasoline costs \(p/gallon and the vehicle gets g miles per gallon. Also assume the driver earns \)w/hour.
e. Should the optimal speed be increased or decreased (compared with part (d)) if L is increased from 400 mi to 500 mi? Explain.
Verified step by step guidance1
First, understand that the total travel cost is composed of two main components: the cost of gasoline and the cost of the driver. The cost of gasoline depends on the price per gallon \( p \) and the vehicle's fuel efficiency \( g \) in miles per gallon. The cost of the driver depends on the wage \( w \) per hour and the time taken to travel the distance \( L \).
The time taken to travel a distance \( L \) at speed \( v \) is given by \( \frac{L}{v} \) hours. Therefore, the cost of the driver is \( w \times \frac{L}{v} \).
The amount of gasoline needed to travel \( L \) miles is \( \frac{L}{g} \) gallons. Thus, the cost of gasoline is \( p \times \frac{L}{g} \).
The total cost \( C \) as a function of speed \( v \) is given by the sum of the gasoline cost and the driver cost: \( C(v) = p \times \frac{L}{g} + w \times \frac{L}{v} \).
To determine whether the optimal speed should be increased or decreased when \( L \) is increased from 400 mi to 500 mi, consider how each component of the cost function \( C(v) \) changes with \( L \). Analyze the derivative of \( C(v) \) with respect to \( v \) to find the optimal speed and observe how it is affected by changes in \( L \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Cost Function
A cost function in this context represents the total expenses incurred during travel, which includes both gasoline costs and driver wages. It can be expressed as a function of distance traveled, speed, and fuel efficiency, allowing for the analysis of how changes in these variables affect overall travel costs.
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Properties of Functions
Marginal Cost
Marginal cost refers to the additional cost incurred by increasing the distance traveled or changing the speed. Understanding marginal costs is crucial for determining the optimal speed, as it helps identify the point where the cost of additional travel equals the benefits gained from reaching the destination faster.
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Optimal Speed
Optimal speed is the speed at which travel costs are minimized for a given distance. It is influenced by factors such as fuel efficiency, time costs associated with the driver's wage, and the distance to be traveled. Analyzing how changes in distance (like from 400 mi to 500 mi) affect optimal speed is essential for making cost-effective travel decisions.
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Related Practice
Textbook Question
First Derivative Test
a. Locate the critical points of f.
b. Use the First Derivative Test to locate the local maximum and minimum values.
c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).
f(x) = x²/(x² - 1) on [-4,4]
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Textbook Question
First Derivative Test
a. Locate the critical points of f.
b. Use the First Derivative Test to locate the local maximum and minimum values.
c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).
f(x) = 2x⁵ - 5x⁴ - 10x³ + 4 on [-2,4]
Textbook Question
Mean Value Theorem The population of a culture of cells grows according to the function P(t) = 100t / t+1, where t ≥ 0 is measured in weeks.
a. What is the average rate of change in the population over the interval [0, 8]?
Textbook Question
First Derivative Test
a. Locate the critical points of f.
b. Use the First Derivative Test to locate the local maximum and minimum values.
c. Identify the absolute maximum and minimum values of the function on the given interval (when they exist).
f(x) = x√(4 - x²) on [-2,2]
Textbook Question
Travel costs A simple model for travel costs involves the cost of gasoline and the cost of a driver. Specifically, assume gasoline costs \(p/gallon and the vehicle gets g miles per gallon. Also assume the driver earns \)w/hour.
b. At what speed does the gas mileage function have its maximum?