Problem 19 - Calculus Early Transcendentals: Pearson New International Edition

Question

Consider that part of the sphere \(x^{2}+y^{2}+z^{2}=a^{2}\) between the planes \(z=h_{1}\) and \(z=h_{2}\), where \(0 \leq h_{1}

Step-by-Step Solution

Answer

The value of \( h \) is \( \frac{h_1 + h_2}{2} \).

Step 1: Understand the Problem

We are tasked with finding a height \( h \) such that the plane \( z = h \) divides the surface area of a spherical cap into two equal parts. The sphere is described by \( x^2 + y^2 + z^2 = a^2 \) and the problem specifies the section of the sphere between two planes \( z = h_1 \) and \( z = h_2 \).

Step 2: Visualize the Sphere Section

Imagine the sphere\( x^2 + y^2 + z^2 = a^2 \) and consider the horizontal slices by the planes \( z = h_1 \) and \( z = h_2 \). The goal is to find a plane \( z = h \) that divides the spherical segment into two parts of equal surface area.

Step 3: Expression for Surface Area of a Spherical Cap

The formula for the surface area of a spherical cap between \( z = c \) and \( z = d \) is given by \( 2\pi a(d - c) \), provided both \( c \) and \( d \) are within the sphere's surface \( -a \le z \le a \).

Step 4: Calculate Total Surface Area of the Spherical Segment

The total surface area of the spherical segment between \( z = h_1 \) and \( z = h_2 \) is calculated as \( 2\pi a(h_2 - h_1) \). This will help in determining the \( h \) such that it divides this area into two equal parts.

Step 5: Solve for \( h \) to Divide Area in Half

For the plane \( z = h \) to divide the surface area into two equal parts, the surface area from \( z = h_1 \) to \( z = h \) must equal the area from \( z = h \) to \( z = h_2 \). Therefore, the condition is: \[ 2\pi a(h - h_1) = 2\pi a(h_2 - h) \] Simplifying, we find: \[ h = \frac{h_1 + h_2}{2} \].

Step 6: Verification

Verify the calculations. By substituting \( h = \frac{h_1 + h_2}{2} \), both parts are \( 2\pi a(h - h_1) \) and \( 2\pi a(h_2 - h) \), confirming both are equal, proving that the surface is indeed split equally.

Key Concepts

Surface Area of a Spherical Cap

A spherical cap is a portion of a sphere that resembles a cap. It is created when a plane slices through a sphere. This concept is common in geometry when exploring various parts of a sphere. The surface area of a spherical cap can be calculated using a specific formula. Consider a sphere with a radius denoted as \( a \). When this sphere is cut by two parallel planes at heights \( z = c \) and \( z = d \), the formula to find the surface area of the cap is: \[ 2\pi a(d - c) \]This formula only applies when both heights are within the surface boundaries of the sphere, i.e., \( -a \leq z \leq a \). The formula essentially calculates the curved surface area between the two planes.When solving problems involving surface areas of spherical caps, it's crucial to correctly identify the radius and the values of \( c \) and \( d \), ensuring that they lie within the sphere's bounds.

Sphere Geometry

Geometry involving spheres is a fascinating area of mathematics. A sphere is defined as the set of points that are all the same distance, called the radius \( a \), from a center point. The equation that represents a sphere in three-dimensional space is: \[ x^2 + y^2 + z^2 = a^2 \]In this context, a spherical cap is a "slice" of this sphere, obtained by slicing through the sphere with a plane.

• The height \( z \) in the equation helps us locate these slices on the sphere.
• The values of \( z \, (h_1, h_2) \) directly influence the size and location of the cap.
• This visualization aids in understanding how certain plane cuts affect the sphere's geometry.

Understanding sphere geometry is essential when calculating areas or volumes of parts of a sphere, such as caps, segments, or sectors.

Calculus Problem Solving in Sphere Geometry

Solving calculus problems involving spheres can often seem challenging, but breaking down the problem can simplify the process. In this scenario, we need to find a particular height \( h \) for which a plane divides a spherical segment into two equal parts by surface area.

• Start by understanding the sphere's equation and visualize the spherical section involved.
• Utilize the surface area formula for the spherical cap to express the problem mathematically.
• Set up an equation to find the desired height \( h \) that divides the area equally.

In the given example, we set the surface areas equal on either side of the plane \( z = h \), resulting in:\[ 2\pi a(h - h_1) = 2\pi a(h_2 - h) \]By solving this simple linear equation, we find:\[ h = \frac{h_1 + h_2}{2} \]This solution demonstrates how calculus and algebra can intersect to solve geometry problems, thus reducing complex problems to manageable calculations.