Question

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 valid range of values for the thickness, \(h\), is \(0 < h \leq 2\).

Step-by-step solution

Substitute the given value of \(r\) into the equation

Since we are asked to find the volume of the cap for a sphere with radius \(1\), we will substitute \(r=1\) into the volume equation \(V=\frac{1}{3} \pi h^{2}(3r-h)\): $$ V=\frac{1}{3} \pi h^{2}(3(1)-h) $$

Simplify the equation

Now let's simplify the equation for the volume: $$ V=\frac{1}{3} \pi h^{2}(3-h) $$

Graph the volume as a function of \(h\)

Now, we will plot the volume \(V\) as a function of thickness \(h\). To do this, you can either use a graphing calculator or an online graphing tool like Desmos or Geogebra. When graphing the function, you'll see it's a cubic function that initially increases and then decreases. It intercepts the \(h\)-axis at \(h=0\) and \(h=3\).

Determine the valid range of \(h\)

By looking at the graph, we can determine the valid range of values for \(h\). We know that \(h\) must be greater than \(0\), since the thickness of the cap cannot be negative. Also, \(h\) cannot be greater than the diameter of the sphere (i.e. \(2\) times the radius), because the cap would no longer be a part of the sphere itself, but rather a whole hemisphere. Therefore, the valid range of \(h\) is: $$ 0 < h \leq 2 $$

Key concepts

Volume of a Cap

The volume of a spherical cap is a fascinating aspect of geometry, and represents the portion of a sphere that is "cap shaped." For a cap with thickness \(h\) on a sphere of radius \(r\), the volume \(V\) can be calculated using the formula: \[ V = \frac{1}{3} \pi h^{2}(3r-h) \]This equation allows us to understand how the thickness of a cap influences its volume.

• As the thickness \(h\) increases, the volume initially increases as well.
• Beyond a certain point, the volume starts decreasing since the formula considers the diminishing space left between \(h\) and \(3r\).

For this problem, we've calculated it for a sphere with a radius of 1, simplifying our calculations and allowing us to focus on how changes in \(h\) affect the volume.

Cubic Function Graph

When graphing the volume of a spherical cap with respect to its thickness on a graph, you are essentially graphing a cubic function. For the specific sphere in this exercise (with radius \(r = 1\)), the graph of the function is:\[ V = \frac{1}{3} \pi h^{2}(3-h) \]This graphic representation highlights the nature of cubic functions:

• The curve of the graph will rise initially until it reaches a peak point, where it will start to fall.
• There are a couple of intercepts on the \(h\)-axis: at \(h = 0\) and \(h = 3\).

Thus, the change of volume with respect to thickness demonstrates how rapidly these values can fluctuate, offering insights into practical applications like manufacturing or design.

Valid Range for Thickness

The range for the thickness \(h\) is a vital component in understanding volume calculations for a spherical cap. Knowing the valid values for \(h\) ensures that the calculations are practical and realistic. The thickness cannot be negative since no physical object can have negative thickness, limiting \(h > 0\). Furthermore, the thickness cannot exceed the diameter, which is twice the sphere's radius. For a unit sphere with radius 1, this means the range for \(h\) must be constrained:

• \(0 < h \leq 2\)

This practical constraint ensures that the cap remains part of the sphere and maintains its geometric integrity.

Sphere Geometry

Sphere geometry is a branch of geometry that deals with the properties and measurements of spheres. A sphere is perfectly symmetrical and all points on its surface are equidistant from its center. This symmetry leads to interesting mathematical properties and equations, such as those derived for volumes of caps, sectors, and more.

• The radius \(r\) is a crucial component in all sphere-related equations, serving as the link between geometric measurements and their mathematical interpretations.
• A spherical cap, as discussed, is formed when a plane slices through a sphere.

The study of sphere geometry, hence, is key to fields such as astronomy, physics, and various engineering disciplines where understanding the shape and volume of spherical objects is crucial.