Problem 62 - Calculus of a Single Variable

Question

Using a Sphere A sphere of radius \(r\) is cut by a plane \(h\) units above the equator, where \(h

Step-by-Step Solution

Answer

The volume of the spherical segment above the plane is given by \(\frac{4}{3}\pi r^{3} - \frac{1}{6}\pi h (3R^{2} + h^{2})\), where \(r\) is the radius of the sphere, \(h\) is the height of the plane above the equator and \(R\) is the radius of the cap as calculated.

Step 1: Calculate the volume of the full sphere

First, calculate the volume of the full sphere with the formula \(\frac{4}{3}\pi r^{3}\). Here, \(r\) is provided in the question as the radius of the sphere.

Step 2: Calculate the radius of the cap

The radius from the cap's center to its outer edge is needed to find its volume. This can be found by Pythagoras' theorem in the right-angled triangle formed by the sphere's center, the point on the sphere's surface directly above the plane cut and the cap's middle. This gives the radius of the cap as \(R = \sqrt{r^{2}-h^{2}}\).

Step 3: Calculate the height of the cap

In the scenario described, the height of the cap will be \(r - h\). This is because the radius from the center of the sphere to its surface is \(r\) and the plane is \(h\) units above the equator.

Step 4: Calculate the volume of the cap

Now that you have the radius and height of the spherical cap, you can find its volume. The formula to find the volume of a spherical cap is \(\frac{1}{6}\pi h (3R^{2} + h^{2})\). Substituting the calculated cap radius and height results in the cap's volume.

Step 5: Subtract the cap's volume from the sphere's

Now that you have both the volume for the complete sphere and the cap alone, subtract the volume of the cap from the volume of the sphere to obtain the volume of the spherical segment above the plane. This will give the final answer.

Key Concepts

Volume Calculation

Calculating volumes in geometry often involves formulas that use the dimensions of the shapes involved. For a sphere, you use the formula \( \frac{4}{3}\pi r^{3} \), where \( r \) is the radius. This formula gives you the total volume of a sphere, a three-dimensional object that is completely round. In problems involving spheres, you may not always need the entire volume. Sometimes, you need just a part of it, like a spherical segment. This requires additional steps, like finding the volume of a spherical cap and then doing some subtraction.

• Calculate the complete sphere's volume: \( \frac{4}{3}\pi r^{3} \).
• Find the volume of the part you need (like a spherical cap).
• Subtract the unwanted volume to find the volume of the segment.

Understanding how to manipulate these formulas allows you to solve complex problems easily.

Spherical Cap

A spherical cap is a portion of a sphere, cut by a plane, that forms a dome-like shape. It can be visualized as the top of a ball sliced off by a flat surface. To find the volume of a spherical cap, you need two key measurements: the radius of the cap and its height. Use the formula \( \frac{1}{6}\pi h (3R^{2} + h^{2}) \) to find the volume of the cap, where \( R \) is its base radius and \( h \) is the height of the cap. Remember the process involves:

• Identifying the base radius \( R \) using the Pythagorean theorem.
• Calculating the cap's height based on how high above the equator the plane cuts the sphere.

This understanding is crucial for finding the volume of a spherical segment.

Geometry

Geometry is all about understanding spaces, shapes, and the properties they have. When you're working with spheres and their parts, like caps or segments, geometry helps determine relationships between different dimensions. The principles of geometry, such as calculating areas and volumes, allow you to solve for missing measurements using known mathematical formulas and theorems. In dealing with spheres:

• Know the sphere's properties, such as the radius.
• Apply geometric formulas to find areas and volumes.
• Use geometric tools like symmetry and plane sections to comprehend shape structure.

These tools and concepts make it easier to visualize and compute the space occupied by spherical objects.

Pythagorean Theorem

The Pythagorean theorem is vital in geometry and is particularly useful when dealing with spheres. It's a fundamental principle that helps in calculating distances in right-angled triangles. According to the theorem, \( a^{2} + b^{2} = c^{2} \), where \( c \) is the hypotenuse's length, opposite the right angle.In the context of finding a spherical cap's dimensions, you can use the theorem to calculate the radius \( R \) of the cap. Consider the triangle formed by:

• The center of the sphere.
• The topmost point on the cap.
• The outer edge of the cap, where the plane cuts the sphere.

The Pythagorean theorem often helps in locating the unknown side of this triangle, facilitating further volume calculations. This approach provides a structured way to solve geometric problems involving spheres.