Question

Volume of a spherical cap A single slice through a sphere of radius \(r\) produces a cap of the sphere. If the thickness of the cap is \(h,\) then its volume is \(V=\frac{1}{3} \pi h^{2}(3 r-h) .\) Graph the volume as a function of \(h\) for a sphere of radius \(1 .\) For what values of \(h\) does this function make sense?

Short answer

Answer: The volume function for the spherical cap of a sphere with radius 1 is given by \(V(h) = \frac{1}{3} \pi h^{2}(3-h)\), with the thickness (\(h\)) ranging from 0 to 2. As the thickness varies within this range, the volume increases and then decreases, reaching its maximum value when h is halfway between 0 and 2.

Step-by-step solution

Determine the function of volume in terms of h

Given the formula \(V=\frac{1}{3} \pi h^{2}(3r-h)\) with \(r=1\), the volume formula becomes: \(V(h) = \frac{1}{3} \pi h^{2}(3-h)\)

Determine the valid domain of h

In order to determine the valid domain for the function \(V(h)\), we consider the geometric constraints of the problem. Since the sphere has a radius of \(1\), the maximum thickness that a cap can have is equivalent to the diameter of the sphere i.e., \(2r = 2\). So, the valid range for \(h\) is: \(0 \leq h \leq 2\)

Graph the function V(h)

To graph the function \(V(h) = \frac{1}{3} \pi h^{2}(3-h)\), we will plot the volume for the valid domain of \(h\). The graph will display the volume of the spherical cap as a function of its thickness for a sphere with radius \(1\). You may use graphing software or a graphing calculator to obtain the graph. The graph will show how the volume of the spherical cap changes as its thickness (\(h\)) varies from \(0\) to \(2\). This will help understand the behavior of the volume function within the valid range of \(h\) values. To conclude, the function \(V(h) = \frac{1}{3} \pi h^{2}(3-h)\) models the volume of a spherical cap with thickness \(h\) and radius \(r=1\). This function makes sense for \(h\) values in the range \(0 \leq h \leq 2\).

Key concepts

Calculus

Understanding the principles of calculus is essential for solving many scientific and mathematical problems, including finding the volume of irregular shapes like a spherical cap. Calculus introduces us to the concept of limits, derivatives, and integrals – powerful tools that allow us to analyze the rates at which things change, and the total accumulation of quantities.

In the context of finding spherical cap volume, calculus helps us to set up and solve equations involving rates of change and accumulated values. Generally, when we derive a formula in calculus, we consider infinitesimally small pieces and sum up (integrate) their contributions to determine the total. For instance, the volume of a shape can often be conceptualized as the sum of innumerable thin slices, each contributing a small amount to the overall volume. When we encounter a formula like the one for a spherical cap volume, we appreciate it as an application of these fundamental calculus concepts.

Volume of Revolution

The volume of revolution is a calculus concept which involves creating a 3D shape by rotating a 2D area around an axis. The resulting shape is symmetrical, and its volume can be found using integral calculus. For example, if you take a semicircle and rotate it around the flat diameter line, you'll get a sphere.

When considering the volume of a spherical cap, you're looking at a portion of a sphere—a shape originally created by revolution. Even though the spherical cap itself isn’t created through revolution, we can still apply principles of rotational volumes to understand its properties. It's interesting to note that the formula provided for the spherical cap volume, \(V=\frac{1}{3} \pi h^{2}(3 r-h)\), is similar in structure to those you’d find when calculating volumes of revolution using the disk method, which involves summing up the volumes of many thin circular disks.

Function Graphing

Graphing functions is a way to visualize how the output of a function changes with different inputs. In this case, we are interested in graphing the volume, \(V\), of a spherical cap as a function of its height, \(h\). By plotting this relationship, we can obtain a clear visual representation of how the volume of a spherical cap varies with changes in \(h\).

Function graphing requires consideration of the domain (valid input values) and the range (possible output values). Here, for a sphere with a fixed radius, we graph volume against varying heights. The curve that emerges from plotting the formula \(V(h) = \frac{1}{3} \pi h^{2}(3-h)\) for a sphere of radius 1 shows us the relationship between \(h\) and \(V\) within the constraints of the problem – a domain of \(h\) from 0 to 2. This visual tool aids in understanding where the function increases, decreases, and what maximal or minimal volumes are achievable.

Domain of a Function

The domain of a function consists of all the input values for which the function is defined. In other words, it's the set of possible values of \(x\) that you can put into the function without returning an error or an undefined result.

Regarding our spherical cap volume function \(V(h)\), the domain is determined by the physical constraints of the problem. Since you cannot have a spherical cap thicker than the sphere itself, and thickness cannot be negative, the domain here for \(h\) is \(0 \leq h \leq 2\) for a sphere of radius 1. Understanding the domain is crucial before graphing the function or integrating it, as it informs us of the values over which the function is valid and thus, should be considered for calculations.