Ch. 4 - Applications of Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 4 - Applications of Derivatives

Problem 4.5.42

Chapter 4, Problem 4.5.42

A cone is formed from a circular piece of material of radius 1 meter by removing a section of angle θ and then joining the two straight edges. Determine the largest possible volume for the cone.

Verified step by step guidance

First, understand that the cone is formed by removing a sector of angle θ from a circle of radius 1 meter. The remaining part of the circle is then shaped into a cone.

The arc length of the removed sector is given by the formula: arc length = θ * radius = θ * 1 = θ. The remaining arc length, which forms the base circumference of the cone, is 2π - θ.

The radius of the base of the cone can be found using the formula for circumference: C = 2πr, where C is the circumference of the base. Thus, r = (2π - θ) / (2π).

The slant height of the cone is the original radius of the circle, which is 1 meter. Use the Pythagorean theorem to find the height of the cone: height = sqrt(1^2 - r^2).

The volume of the cone is given by the formula: V = (1/3)πr^2h. Substitute the expressions for r and h into this formula to express the volume in terms of θ, and then find the value of θ that maximizes the volume.

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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 can be calculated using the formula V = (1/3)πr²h, where r is the radius of the base and h is the height of the cone. Understanding this formula is crucial for determining the maximum volume of the cone formed from the circular piece of material.

Geometry of the Cone

When a sector of a circle is removed to form a cone, the radius of the original circle becomes the slant height of the cone. The angle θ determines the dimensions of the cone, including its base radius and height, which are essential for calculating the volume.

Optimization in Calculus

Optimization involves finding the maximum or minimum values of a function. In this context, we need to apply calculus techniques, such as taking derivatives and setting them to zero, to find the angle θ that maximizes the volume of the cone formed from the circular material.

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