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Chapter 7: Problem 64

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

Short Answer

Expert verified
The volume of the spherical segment can be obtained by subtracting the volume of the small sphere \(V_{small} = \frac{4}{3}\pi r'^3\) from the volume of the original sphere \(V_{large} = \frac{4}{3}\pi r^3\). The volume of the spherical segment will be \(V_{segment} = V_{large} - V_{small}\).

Step by step solution

01

Sketch the problem

For better understanding of what is asked, imagine the sketch of a sphere cut by a plane. The result will be a smaller sphere located on top of the original sphere's equator.
02

Calculate the volume of the small sphere

Calculate the radius of the small sphere. The radius \(r'\) of the small sphere is equal to \(r - h\). As we know that the volume \(V\) of sphere is calculated by the formula \(V = \frac{4}{3}\pi r^3\), apply this formula using the radius \(r'\) to get the volume of the small sphere.
03

Calculate the volume of the spherical segment

The volume of the spherical segment is the difference between the volumes of the large sphere and the small sphere. Subtract the volume of the small sphere from the volume of the original sphere to get the volume of the spherical segment.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Sphere
A sphere is a perfectly round three-dimensional shape, similar to a ball. Each point on the surface of a sphere is at an equal distance from its center. This center point is crucial in defining the sphere and its properties.
If we look at everyday objects, a basketball or a globe are both great examples of spheres.
  • Key characteristic: All surface points are equidistant from the center.
  • Used in many applications: physics, engineering, and computer graphics.
Understanding spheres is essential in geometry since they are one of the fundamental solid shapes. They are symmetrically round and have fascinating properties, which are vital for forming spherical structures.
Exploring Volume
Volume measures the space occupied by a three-dimensional object. For spheres, the formula to calculate volume is \[ V = \frac{4}{3} \pi r^3 \] where \(r\) is the radius of the sphere. This formula helps to find how much "space" the sphere contains inside its boundaries.
  • Important for figuring out how much liquid or gas a sphere can hold.
  • Helps in understanding capacity and storage volume in real-world applications.
The concept of volume is widely used not only for spheres but also for various shapes, helping us in areas like construction, packaging, and science.
Decoding Radius
The radius of a sphere is the distance from its center to any point on the surface. It's a crucial measurement because it directly influences the sphere鈥檚 size and volume. The radius is half of the sphere鈥檚 diameter, the longest line segment that passes through the center and touches two points on the surface.
  • Radius is essential for calculating volume and surface area.
  • Basic feature that defines the sphere's size.
In equations, you鈥檒l often see a radius squared or cubed, which emphasizes its importance in geometric and physical calculations. Whether building, designing, or exploring mathematical concepts, understanding the radius is central to working with spheres.
Geometry in Spheres
Geometry is the branch of mathematics dealing with shapes, sizes, and the properties of space. In the context of spheres, geometry helps us understand how these symmetrical shapes interact with other objects and spaces. For a sphere, aspects like radius, diameter, and volume are critical geometric elements.
  • Geometry explains the relationships and properties of parts of a sphere.
  • Helps in solving problems involving intersections with planes or other shapes.
Studying geometry provides tools to calculate various properties and measurements, enabling practical solutions to geometric problems, like finding the volume of spherical segments. The geometric understanding of spheres is crucial for fields such as architecture, astronomy, and more.

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Most popular questions from this chapter

The hydraulic cylinder on a woodsplitter has a 4 -inch bore (diameter) and a stroke of 2 feet. The hydraulic pump creates a maximum pressure of 2000 pounds per square inch. Therefore, the maximum force created by the cylinder is \(2000\left(\pi \cdot 2^{2}\right)=8000 \pi\) pounds. (a) Find the work done through one extension of the cylinder, given that the maximum force is required. (b) The force exerted in splitting a piece of wood is variable. Measurements of the force obtained in splitting a piece of wood are shown in the table. The variable \(x\) measures the extension of the cylinder in feet, and \(F\) is the force in pounds. Use the regression capabilities of a graphing utility to find a fourth-degree polynomial model for the data. Plot the data and graph the model. $$ \begin{array}{|l|l|c|c|c|c|c|c|} \hline x & 0 & \frac{1}{3} & \frac{2}{3} & 1 & \frac{4}{3} & \frac{5}{3} & 2 \\\ \hline F(x) & 0 & 20,000 & 22,000 & 15,000 & 10,000 & 5000 & 0 \\ \hline \end{array} $$ (c) Use the model in part (b) to approximate the extension of the cylinder when the force is maximum. (d) Use the model in part (b) to approximate the work done in splitting the piece of wood.

Newton's Law of Universal Gravitation Consider two particles of masses \(m_{1}\) and \(m_{2} .\) The position of the first particle is fixed, and the distance between the particles is \(a\) units. Using Newton's Law of Universal Gravitation, find the work needed to move the second particle so that the distance between the particles increases to \(b\) units.

In Exercises 43-46, the integral represents the volume of a solid of revolution. Identify (a) the plane region that is revolved and (b) the axis of revolution. $$2 \pi \int_{0}^{1}(4-x) e^{x} d x$$

In Exercises 3-6, the area of the top side of a piece of sheet metal is given. The sheet metal is submerged horizontally in 8 feet of water. Find the fluid force on the top side. 25 square feet

Approximation In Exercises 31 and \(32,\) approximate the are length of the graph of the function over the interval \([0,4]\) in three ways.(a) Use the Distance Formula to find the distance between the endpoints of the arc. (b) Use the Distance Formula to find the lengths of the four line segments connecting the points on the arc when \(x=0, x=1, x=2, x=3,\) and \(x=4\) .Find the sum of the four lengths. (c) Use the integration capabilities of a graphing utility to approximate the integral yielding the indicated are length. $$f(x)=x^{3}$$
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