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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblema 4.R.39
Capitolo 4, Problema 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>

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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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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.
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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.
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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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