Problem 46

Question

For what value of $c$ is the volume of the ellipsoid $x^{2}+(y / 2)^{2}+(z / c)^{2}=1$ equal to $8 \pi ?$

Step-by-Step Solution

Verified

Answer

The value of $c$ is 3.

1Step 1: Understand the Ellipsoid Equation

The given equation is $x^{2}+(y/2)^{2}+(z/c)^{2}=1$, which represents an ellipsoid centered at the origin with semi-axes lengths 1, 2, and $c$ along the x, y, and z directions, respectively.

2Step 2: Volume Formula for an Ellipsoid

The volume $V$ of an ellipsoid with semi-axes $a$, $b$, and $c$ is given by the formula $V = \frac{4}{3} \pi a b c$.

3Step 3: Substitute the Semi-Axes Lengths

For this ellipsoid, the semi-axes are $a = 1$, $b = 2$, and $c = c$. Substitute these into the volume formula to get $V = \frac{4}{3} \pi (1) (2) (c) = \frac{8}{3} \pi c$.

4Step 4: Set Volume Equal to Given Value

Set the volume expression equal to the given volume $8 \pi$ to form the equation: $\frac{8}{3} \pi c = 8 \pi$.

5Step 5: Solve for $c$

Divide both sides by $\pi$, resulting in $\frac{8}{3} c = 8$. Solve for $c$ by multiplying both sides by $\frac{3}{8}$, giving $c = 3$.

Key Concepts

Ellipsoid EquationVolume FormulaSemi-Axes

Ellipsoid Equation

An ellipsoid is a three-dimensional geometric shape that resembles a stretched or compressed sphere. The equation used to depict an ellipsoid in a coordinate system is quite similar to that of a sphere but includes different denominators for each variable to represent its unique dimensions. The standard form of an ellipsoid equation is: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1 \] This equation is quite versatile as it describes how the ellipsoid stretches along the x, y, and z axes. It's important to note:

The center of the ellipsoid is at the origin (0, 0, 0).
The semi-axes lengths are $a$, $b$, and $c$, respectively.
This form helps in visualizing the spread of the ellipsoid in different directions.

In the given problem, the equation takes the form $x^{2}+(y/2)^{2}+(z/c)^{2}=1$. This implies the semi-axes lengths are 1, 2, and $c$ for the x, y, and z dimensions, respectively. The form matches the standard ellipsoid equation, indicating an elongation along specific axes, determined by the denominators.

Volume Formula

Calculating the volume of an ellipsoid is a fascinating extension of the formula for a sphere's volume. Unlike a sphere that has the same radius in all directions, an ellipsoid has different semi-axis lengths along each coordinate direction. Therefore, its volume can be computed using the following formula:\[ V = \frac{4}{3} \pi a b c \]Here's a quick explanation of each component in the formula:

$\frac{4}{3}\pi$: This fraction is derived from the volume formula of a sphere, accounting for the three-dimensional spread of the shape.
$a$, $b$, and $c$: These variables represent the semi-axes lengths of the ellipsoid in the x, y, and z directions.

For our exercise, the semi-axes are $a = 1$, $b = 2$, and $c = c$. Substituting these values into the volume formula results in:\[ V = \frac{4}{3} \pi (1) (2) (c) = \frac{8}{3} \pi c \]This tells us how the size of the ellipsoid changes with its axis dimensions. Setting this expression equal to the required volume simplifies finding the missing value for $c$.

Semi-Axes

The semi-axes of an ellipsoid are fundamental to understanding its overall shape and orientation. These semi-axes are, essentially, half the lengths of the ellipsoid along each principal axis. In simpler terms, they are the radii of the ellipsoid in the x, y, and z directions:

Semi-major axis: The longest axis of the ellipsoid.
Semi-minor axis: The shortest axis of the ellipsoid.
Intermediate axis: An axis that is neither the longest nor the shortest.

In the context of our given ellipsoid equation, $x^{2}+(y/2)^{2}+(z/c)^{2}=1$, the semi-axes are specified as:

$a = 1$
$b = 2$
$c$, which needs to be determined based on the volume requirement.

These semi-axes tell us that the ellipsoid stretches further in the y direction than in the x or z directions, assuming $c$ is not larger than 2. Understanding these axial lengths is key to visualizing the orientation and the extent of an ellipsoid in three-dimensional space.

Problem 45

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