Chapter 8: Problem 74
A portion of a sphere of radius \(r\) is removed by cutting out a circular cone with its vertex at the center of the sphere. The vertex of the cone forms an angle of \(2 \theta .\) Find the surface area removed from the sphere.
Short Answer
Expert verified
The surface area removed from the sphere is given by \(\pi r^2 (3 - \sin(\theta) - \sec(\theta))\).
Step by step solution
01
Identify Given Values
The radius of the sphere is given as \(r\) and the vertex angle of the cone cut out from the sphere is \(2\theta\).
02
Determine the Surface Area of the Sphere
The total surface area of the sphere is given by the formula \(4\pi r^2\).
03
Calculate the Cone Base Radius and Slant Height
The height of the cone will be equal to the radius of the sphere and can be denoted as \(r\). The radius of the cone's base can be found using trigonometric relations, giving \(r \sin(\theta)\), and the slant height of the cone comes out as \(r\sec(\theta)\) using Pythagoras theorem.
04
Determine the Surface Area of the Cone
The total surface area of a cone is given by the formula \(\pi r l + \pi r^2\) where \(r\) is the radius of the base and \(l\) is the slant height. Substituting the derived values for the radius and slant height from step 3 we get, \(\pi r^2 \sin(\theta) + \pi r^2 \sec(\theta)\).
05
Calculate the Surface Area Removed from the Sphere
The surface area removed from the sphere is equal to the difference between the surface area of the sphere and the surface area of the cone. Therefore, \((4\pi r^2) - (\pi r^2 \sin(\theta) + \pi r^2 \sec(\theta)) = \pi r^2 (3 - \sin(\theta) - \sec(\theta))\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Spherical Sector
When a sphere has a portion removed by cutting out a cone with its vertex at the center of the sphere, the resulting shape is called a spherical sector. This sector contains both a curved surface and a flat base, akin to a spherical cap. To understand the concept, visualize removing a scoop of ice cream from a perfectly round scoop.
Calculating the surface area or the volume of a spherical sector involves understanding the geometric properties of spheres and cones. In the exercise given, the spherical sector is essentially what remains of a sphere with radius \( r \) after a cone is removed. The surface area removed is part of the surface area of the spherical sector. Trigonometry often plays a role in such calculations, especially when dealing with the pointed segment of the sector which is the cone. By mastering the geometry of both the sphere and cone, one can adeptly solve problems related to spherical sectors.
Calculating the surface area or the volume of a spherical sector involves understanding the geometric properties of spheres and cones. In the exercise given, the spherical sector is essentially what remains of a sphere with radius \( r \) after a cone is removed. The surface area removed is part of the surface area of the spherical sector. Trigonometry often plays a role in such calculations, especially when dealing with the pointed segment of the sector which is the cone. By mastering the geometry of both the sphere and cone, one can adeptly solve problems related to spherical sectors.
Cone Volume and Surface Area
Cones are three-dimensional shapes with a circular base and a single vertex. To calculate the volume of a cone, you use the formula \( V = \frac{1}{3}\pi r^2h \), where \( r \) is the radius of the base, and \( h \) is the height of the cone.
For surface area, however, we must consider both the base and the curved surface (also known as the lateral surface). The latter can be calculated using the formula for the lateral surface area of a cone, \( A = \pi rl \), where \( l \) is the slant height. The slant height can be derived using the Pythagorean theorem, as it is the hypotenuse of the triangle formed by the height of the cone and the radius of the base when the cone is bisected. The total surface area is the sum of the base and lateral surface areas, given by \( A = \pi r(l + r) \).
Understanding these components is essential when solving problems related to the spheres and cones as they often involve dissections or combinations of such shapes.
For surface area, however, we must consider both the base and the curved surface (also known as the lateral surface). The latter can be calculated using the formula for the lateral surface area of a cone, \( A = \pi rl \), where \( l \) is the slant height. The slant height can be derived using the Pythagorean theorem, as it is the hypotenuse of the triangle formed by the height of the cone and the radius of the base when the cone is bisected. The total surface area is the sum of the base and lateral surface areas, given by \( A = \pi r(l + r) \).
Understanding these components is essential when solving problems related to the spheres and cones as they often involve dissections or combinations of such shapes.
Trigonometric Relationships
Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles. The fundamental trigonometric functions—sine (sin), cosine (cos), and tangent (tan)—can be used to solve problems involving right-angled triangles.
In our exercise, for example, trigonometric relationships are applied to find the radius of the cone's base using the angle \( \theta \) and the sphere's radius. The sine function relates the base radius \( r \sin(\theta) \) to the sphere's radius by opposite over hypotenuse. Moreover, the secant function (reciprocal of cosine) helps in determining the slant height of the cone, \( r\sec(\theta) \), which is crucial when calculating the cone's surface area. Mastering the relationships between these trigonometric functions and right-angled triangles allows us to navigate the geometry involved in three-dimensional figures like cones and spheres.
In our exercise, for example, trigonometric relationships are applied to find the radius of the cone's base using the angle \( \theta \) and the sphere's radius. The sine function relates the base radius \( r \sin(\theta) \) to the sphere's radius by opposite over hypotenuse. Moreover, the secant function (reciprocal of cosine) helps in determining the slant height of the cone, \( r\sec(\theta) \), which is crucial when calculating the cone's surface area. Mastering the relationships between these trigonometric functions and right-angled triangles allows us to navigate the geometry involved in three-dimensional figures like cones and spheres.
Pythagorean Theorem
The Pythagorean theorem is a cornerstone of geometry, stating that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be expressed algebraically as \( a^2 + b^2 = c^2 \).
In the context of our spherical sector problem, we utilize the Pythagorean theorem to determine the slant height of the cone. Given the spherical radius \( r \) as the height of the cone and also as the hypotenuse of the right-angled triangle formed by the cone's dimensions, we find the slant height to be \( r\sec(\theta) \). By applying the theorem, we validate that the square of the sphere's radius is equal to the sum of the squares of the cone's base radius and height. This connection between trigonometry and geometry is fundamental in solving a wide range of problems involving three-dimensional shapes.
In the context of our spherical sector problem, we utilize the Pythagorean theorem to determine the slant height of the cone. Given the spherical radius \( r \) as the height of the cone and also as the hypotenuse of the right-angled triangle formed by the cone's dimensions, we find the slant height to be \( r\sec(\theta) \). By applying the theorem, we validate that the square of the sphere's radius is equal to the sum of the squares of the cone's base radius and height. This connection between trigonometry and geometry is fundamental in solving a wide range of problems involving three-dimensional shapes.