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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 4.5.36
Chapter 4, Problem 4.5.36

Covering a marble Imagine a flat-bottomed cylindrical pot with a circular cross section of radius 4. A marble with radius 0 < r < 4 is placed in the bottom of the pot. What is the radius of the marble that requires the most water to cover it completely?

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Understand the problem: We need to find the radius of the marble that requires the most water to cover it completely when placed at the bottom of a cylindrical pot with a radius of 4.
Visualize the situation: The marble is a sphere with radius r, and it is placed at the bottom of the cylinder. The water will form a spherical cap over the marble.
Determine the volume of water needed: The volume of water required to cover the marble is the volume of the spherical cap. The formula for the volume of a spherical cap is \( V = \frac{1}{3} \pi h^2 (3R - h) \), where h is the height of the cap and R is the radius of the sphere.
Relate the height of the cap to the radius of the marble: The height h of the cap is related to the radius of the marble r by the formula \( h = R - \sqrt{R^2 - r^2} \). Substitute this into the volume formula.
Maximize the volume: To find the radius r that requires the most water, take the derivative of the volume with respect to r, set it equal to zero, and solve for r. This will give the critical points, and you can determine which one 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 Cylinder

The volume of a cylinder is calculated using the formula V = πr²h, where r is the radius of the base and h is the height. In this context, the cylindrical pot's volume will help determine how much water is needed to cover the marble. Understanding this formula is essential for calculating the water volume required based on the marble's radius.
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Volume of a Sphere

The volume of a sphere is given by the formula V = (4/3)πr³, where r is the radius of the sphere. This concept is crucial for determining the volume of the marble, which directly affects how much water is needed to cover it. Knowing how to apply this formula allows for the calculation of the marble's volume based on its radius.
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Optimization

Optimization in calculus involves finding the maximum or minimum values of a function. In this problem, we need to determine the radius of the marble that maximizes the volume of water needed to cover it. This requires setting up a function that relates the marble's radius to the water volume and using techniques such as differentiation to find the optimal radius.
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