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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 48
Chapter 4, Problem 48

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.

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Start by understanding the geometry of the problem: A right circular cylinder is inscribed in a sphere. The sphere has a fixed radius R, and the cylinder's height h and radius r are variables that need to be optimized for maximum volume.
Express the relationship between the cylinder and the sphere using the Pythagorean theorem. The diagonal of the cylinder (which is the diameter of the sphere) can be expressed as: h2 + 4r2 = 2R.
The volume V of the cylinder is given by the formula: V = πr2h. Substitute the expression for h from the Pythagorean relationship into the volume formula to express V in terms of r alone.
Differentiate the volume function with respect to r to find the critical points. This involves using the chain rule and simplifying the derivative to find where it equals zero, indicating potential maximum volume.
Analyze the critical points and use the second derivative test or other methods to confirm which point gives the maximum volume. Ensure that the values of h and r satisfy the original geometric constraints of the problem.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Right Circular Cylinder

A right circular cylinder is a three-dimensional geometric shape with two parallel circular bases connected by a curved surface at a fixed distance from the center. The height (h) is the distance between the bases, and the radius (r) is the radius of the circular bases. Understanding the properties of this shape is essential for calculating its volume and optimizing dimensions.
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Volume of a Cylinder

The volume (V) of a right circular cylinder is calculated using the formula V = πr²h, where r is the radius of the base and h is the height. This formula is crucial for determining the maximum volume of the cylinder inscribed within a sphere, as it directly relates the dimensions of the cylinder to its capacity.
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Inscribed Shapes

An inscribed shape is one that fits perfectly within another shape, touching it at certain points. In this case, the right circular cylinder is inscribed in a sphere of radius R, meaning that the cylinder's dimensions must be constrained by the sphere's radius. This relationship is key to solving the problem, as it involves using geometric properties and optimization techniques to find the maximum volume.
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Real World Application
Related Practice
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.


e. Should the optimal speed be increased or decreased (compared with part (d)) if L is increased from 400 mi to 500 mi? Explain.

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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) = x² + 3 on [-3,2]

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Textbook Question

Suppose you own a tour bus and you book groups of 20 to 70 people for a day tour. The cost per person is \$30 minus \$0.25 for every ticket sold. If gas and other miscellaneous costs are \$200, how many tickets should you sell to maximize your profit? Treat the number of tickets as a nonnegative real number.

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

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?