Problem 38 - Calculus Early Transcendentals: Pearson New International Edition

Question

A right circular cylinder of radius \(r\) is inscribed in a sphere of radius \(2 r\). Find a formula for \(V(r)\), the volume of the cylinder, in terms of \(r\).

Step-by-Step Solution

Answer

The volume formula is \( V(r) = 2\pi\sqrt{3} r^3 \).

Step 1: Identify the Sphere

The problem states that a cylinder is inscribed in a sphere. The sphere has a radius of \(2r\). This is important because it determines the constraints for the dimensions of the cylinder.

Step 2: Understand Cylinder and Sphere Relationship

Since the cylinder is inscribed in the sphere, the diagonal of the cylinder is equal to the diameter of the sphere. Thus, the diagonal of the cylinder is \(4r\) (since the diameter of a sphere is twice its radius \(2 \times 2r = 4r\)).

Step 3: Calculate the Height of the Cylinder

The cylinder's diagonal forms the hypotenuse of a right triangle with the height \(h\) of the cylinder as one leg and \(2r\) (double the radius of the circular base) as the other leg. Use the Pythagorean theorem: \[ h^2 + (2r)^2 = (4r)^2. \] Solving for \(h\), \( h^2 = 16r^2 - 4r^2 = 12r^2 \), so \( h = \sqrt{12r^2} = 2\sqrt{3}r \).

Step 4: Compute Volume of the Cylinder

The formula for the volume of a cylinder is \( V = \pi r^2 h \). Substituting \( h = 2\sqrt{3}r \), we find \[ V = \pi r^2 (2\sqrt{3}r) = 2\pi\sqrt{3} r^3. \] Thus, the volume of the cylinder in terms of \(r\) is \( V(r) = 2\pi\sqrt{3} r^3 \).

Key Concepts

Right Circular Cylinder

A right circular cylinder is a 3D geometric shape with a circular base and a specific height. Unlike other cylinders, the sides of a right circular cylinder are perpendicular to the bases. This creates a clear distinction between this type of cylinder and slanted ones.
Understanding the right circular cylinder is crucial as it is a key component in many geometry problems. The volume of a cylinder is calculated using the formula

• Volume = π × radius² × height

This formula highlights that both the radius of the base and the height of the cylinder are integral in determining its volume. The right circular cylinder is often seen in practical scenarios, offering a perfect introduction to more complex geometry topics.

Sphere Geometry

Sphere geometry involves the study of spheres, which are perfectly symmetrical 3D shapes. These shapes are defined by one essential measurement: the radius. The radius is the distance from the center of the sphere to any point on its surface.
This concept is crucial in our problem where a cylinder is inscribed into a sphere with a given radius, forming a symmetric relationship with it. Key points about spheres include:

• The diameter is twice the radius.
• All points on a sphere's surface are equidistant from the center.
• Spheres have no edges or vertices.

Understanding these properties helps to visualize the problem and sets up the calculation of the inscribed shape's dimensions.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that relates the lengths of the three sides of a right triangle. It states that for a right triangle with legs of lengths 'a' and 'b', and hypotenuse 'c', the relationship is

• \[ a^2 + b^2 = c^2 \]

In the context of our cylinder problem, the Pythagorean theorem is used to find the height of the cylinder.
By treating the diagonal of the cylinder as the hypotenuse of a right triangle, we combine the base and height dimensions to solve for the unknown value.This approach is particularly useful in situations involving circular objects inscribed within another shape, as it allows for precise calculations based on fixed and known measurements.

Cylinder Height Calculation

Calculating the height of a cylinder when it is inscribed in a sphere involves using known geometric principles. The diagonal length of the cylinder's base serves as a crucial aspect here, forming one leg of a right triangle. The other leg is the cylinder's height.We use the formula from the Pythagorean theorem:

• \[ h^2 + (2r)^2 = (4r)^2 \]

Here, the diagonal, which equals the sphere's diameter, is the hypotenuse. Solving for 'h',

• \[ h = \sqrt{12r^2} = 2\sqrt{3}r \]

This calculated height is essential for determining the cylinder's volume, providing a clear basis from geometrical principles to practical application in volume calculation. Using these steps ensures accuracy and understanding in solving similar spherical problems.