Ch. 4 - Applications of the Derivative

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 4 - Applications of the Derivative

Problem 4.R.39

Chapter 4, Problem 4.R.39

Optimization A right triangle has legs of length h and r and a hypotenuse of length 4 (see figure). It is revolved about the leg of length h to sweep out a right circular cone. What values of h and r maximize the volume of the cone? (Volume of a cone = (1/3) πr²h.) <IMAGE>

Verified step by step guidance

First, express the relationship between the sides of the right triangle using the Pythagorean theorem: h² + r² = 4².

Solve the Pythagorean equation for one of the variables, say r, in terms of h: r = sqrt(16 - h²).

Substitute the expression for r into the volume formula of the cone: V = (1/3)π(sqrt(16 - h²))²h.

Simplify the volume expression: V = (1/3)π(16 - h²)h.

To find the maximum volume, take the derivative of V with respect to h, set it equal to zero, and solve for h. Then, use this value to find r using the expression from step 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 Cone

The volume of a cone is calculated using the formula V = (1/3)πr²h, where r is the radius of the base and h is the height. This formula derives from the geometric properties of cones and is essential for determining the maximum volume in optimization problems. Understanding how to manipulate this formula is crucial for solving the given problem.

Optimization

Optimization in calculus involves finding the maximum or minimum values of a function. In this context, we need to maximize the volume of the cone by adjusting the dimensions h and r. Techniques such as taking derivatives and applying critical point analysis are fundamental to solving optimization problems effectively.

Right Triangle Relationships

In this problem, the right triangle formed by the legs h and r and the hypotenuse (length 4) is governed by the Pythagorean theorem, which states that a² + b² = c². This relationship allows us to express one variable in terms of the other, facilitating the optimization process by reducing the number of variables involved.

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Related Practice

Textbook Question

104–107. Functions from derivatives Find the function f with the following properties.

h'(x) = (x⁴ -2) /(1 + x²) ; h (1) = -(2/3)

Textbook Question

Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).

ƒ(x) = 4x¹⸍² - x⁵⸍² on [0, 4]

Textbook Question

82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.

2ˣ and 4ˣ⸍²

Textbook Question

24–34. Curve sketching Use the guidelines given in Section 4.4 to make a complete graph of the following functions on their domains or on the given interval. Use a graphing utility to check your work.

{Use of Tech} ƒ(x) = x (x -1)e⁻ˣ

Textbook Question

Use the graphs of ƒ' and ƒ" to complete the following steps. <IMAGE>

b. Determine the locations of the inflection points of f and the intervals on which f is concave up or concave down.

Textbook Question

82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.

eˣ and 3ˣ