Problem 27 - Calculus Early Transcendentals: Pearson New International Edition

Question

What are the dimensions of the right circular cylinder with greatest curved surface area that can be inscribed in a sphere of radius \(r\) ?

Step-by-Step Solution

Answer

The cylinder's dimensions are diameter \(d = \sqrt{2}r\) and height \(h = r\sqrt{2}\).

Step 1: Understand the Sphere and Cylinder Relationship

The cylinder is inscribed in a sphere means that the diameter of the sphere is equal to the diagonal of the cylinder's rectangular cross-section. The diagonal can be found using the Pythagorean theorem in terms of the cylinder's height \(h\) and diameter \(d\).

Step 2: Set Up the Geometric Relationship

Since the sphere radius is \(r\), the diameter is \(2r\). For the inscribed cylinder, using the relationship \((\frac{d}{2})^2 + (\frac{h}{2})^2 = r^2\) ensures the cylinder fits inside the sphere.

Step 3: Express Curved Surface Area of Cylinder

The curved surface area (CSA) of a cylinder is calculated using \(2\pi rh\) where \(r\) is the base radius of the cylinder and \(h\) is its height. In terms of base diameter \(d\), \(r = \frac{d}{2}\), so CSA becomes \(\pi dh\).

Step 4: Use Constraint to Express Height in Terms of Diameter

From Step 2, \((\frac{d}{2})^2 + (\frac{h}{2})^2 = r^2\) simplifies to \(d^2 + h^2 = 4r^2\), allowing us to solve for \(h = \sqrt{4r^2 - d^2}\).

Step 5: Substitute for \(h\) in CSA Formula

Substitute \(h = \sqrt{4r^2-d^2}\) into CSA formula, resulting in \(CSA = \pi d \sqrt{4r^2 - d^2}\).

Step 6: Find the Maximum CSA Using Calculus

To maximize, take the derivative of \(CSA\) with respect to \(d\) and set it to zero. Solving \( \frac{d}{dd} \pi d \sqrt{4r^2-d^2} = 0 \), leads to the optimal diameter found as \(d = \sqrt{2}r\).

Step 7: Calculate Optimal Height

Once we have \(d = \sqrt{2}r\), use this in \(h = \sqrt{4r^2 - d^2}\) to find \(h = \sqrt{4r^2 - 2r^2} = r\sqrt{2}\).

Key Concepts

Curved Surface Area

To understand curved surface area, think about just the side part of a can without its top or bottom. For a cylinder, the curved surface area (CSA) can be calculated using the formula: \( 2\pi rh \), where \( r \) is the base radius, and \( h \) is the height of the cylinder. However, when expressed in terms of the diameter \( d \), which defines the base, we use \( r = \frac{d}{2} \). The formula thus becomes \( \pi dh \). This formula represents the area you would get if you rolled out the side of the cylinder into a flat rectangle. Understanding this concept is crucial when optimizing shapes to fit within others, like our cylinder within the sphere.

Pythagorean Theorem

The Pythagorean theorem is essential in solving geometric problems involving right triangles. It says that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. For a cylinder inscribed in a sphere, we relate this theorem to the cylinder's diagonal and the sphere's diameter. Specifically, for the cylinder, the diagonal of its cross-section forms a hypotenuse with its height \( h \) and diameter \( d \). The relationship \( (\frac{d}{2})^2 + (\frac{h}{2})^2 = r^2 \) ensures that the cylinder fits perfectly within the sphere of radius \( r \). The squared terms reflect the connection between the cylinder's dimensions under the constraint imposed by the sphere.

Calculus Optimization

Optimization in calculus often involves finding maximum or minimum values of a function. When we want the cylinder to have the greatest curved surface area while being inscribed in a sphere, we use calculus optimization. We start by expressing the CSA as a function of \( d \), the diameter, that is \( \pi d \sqrt{4r^2 - d^2} \). To find the maximum, take the derivative of the CSA function concerning \( d \), and set this derivative equal to zero. Solving \( \frac{d}{dd} \pi d \sqrt{4r^2-d^2} = 0 \) gives the optimal diameter \( d = \sqrt{2}r \). This process identifies the point at which the rate of change of the CSA with respect to \( d \) is zero, indicating either a maximum or minimum. In this case, it tells us the maximum CSA obtainable for our cylinder.

Geometric Relationships

Understanding geometric relationships helps piece together how different dimensions interact within multiple shapes. For the cylinder and sphere, the sphere's diameter is the major constraint governing these interactions. The relationship \( d^2 + h^2 = 4r^2 \) is derived by combining the Pythagorean theorem with the sphere's control over the cylinder's dimensions. Here, \( 2r \) is the diameter of the sphere, limiting the cylinder's overall outward stretch to fit inside. This equation allows us to express one dimension in terms of another, e.g., \( h = \sqrt{4r^2 - d^2} \). By using this relationship, the cylinder's dimensions can be modified or optimized while respecting the sphere's boundary.