Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. F(x) = x³ - 4x + 100 and G(x) = x³ - 4x - 100 are antiderivatives of the same function.
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 4.5.35.a
Optimal soda can
a. Classical problem Find the radius and height of a cylindrical soda can with a volume of 354 cm³ that minimize the surface area.
Verified step by step guidance1
Start by identifying the formulas involved: The volume of a cylinder is given by \( V = \pi r^2 h \) and the surface area is given by \( A = 2\pi r^2 + 2\pi rh \).
Given that the volume \( V = 354 \text{ cm}^3 \), express the height \( h \) in terms of the radius \( r \) using the volume formula: \( h = \frac{354}{\pi r^2} \).
Substitute \( h \) from the volume equation into the surface area formula to express the surface area \( A \) solely in terms of \( r \): \( A = 2\pi r^2 + \frac{708}{r} \).
To find the radius that minimizes the surface area, take the derivative of \( A \) with respect to \( r \), set it equal to zero, and solve for \( r \). This involves finding \( \frac{dA}{dr} = 4\pi r - \frac{708}{r^2} \) and setting \( \frac{dA}{dr} = 0 \).
Solve the equation \( 4\pi r - \frac{708}{r^2} = 0 \) to find the critical points. Then, use the second derivative test or analyze the behavior of \( \frac{dA}{dr} \) to confirm that the solution gives a minimum surface area. Finally, use the value of \( r \) to find \( h \) using the expression \( h = \frac{354}{\pi r^2} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Volume of a Cylinder
The volume of a cylinder is calculated using the formula V = πr²h, where r is the radius and h is the height. In this problem, the volume is given as 354 cm³, which serves as a constraint for determining the optimal dimensions of the can.
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Surface Area of a Cylinder
The surface area of a cylinder is given by the formula A = 2πr² + 2πrh, where the first term accounts for the areas of the two circular bases and the second term accounts for the lateral surface area. Minimizing this surface area while maintaining a fixed volume is the core objective of the problem.
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Optimization Techniques
Optimization in calculus involves finding the maximum or minimum values of a function. In this case, techniques such as setting up a function for surface area in terms of one variable (using the volume constraint) and applying derivatives to find critical points are essential for solving the problem.
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Related Practice
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The function f(x) = √x has a local maximum on the interval [0,∞).
Textbook Question
Maximizing profit Suppose a tour guide has a bus that holds a maximum of 100 people. Assume his profit (in dollars) for taking people on a city tour is P(n) = n(50 - 0.5n) - 100. (Although P is defined only for positive integers, treat it as a continuous function.)
a. How many people should the guide take on a tour to maximize the profit?
Textbook Question
{Use of Tech} Elliptic curves The equation y² = x³ - ax + 3, where a is a parameter, defines a well-known family of elliptic curves.
a. Plot a graph of the curve when a = 3.
Textbook Question
107–110. {Use of Tech} Motion with gravity Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation a(t) = v' (t) = -g , where g = 9.8 m/s² .
a. Find the velocity of the object for all relevant times.
A payload is released at an elevation of 400 m from a hot-air balloon that is rising at a rate of 10 m/s.
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Textbook Question
{Use of Tech} Fixed points of quadratics and quartics Let f(x) = ax(1 -x), where a is a real number and 0 ≤ a ≤ 1. Recall that the fixed point of a function is a value of x such that f(x) = x (Exercises 48–51).
a. Without using a calculator, find the values of a, with 0 ≤ a ≤ 4, such that f has a fixed point. Give the fixed point in terms of a.