Problem 46 - Thomas Calculus

Question

Elipsoid 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

Answer

The value of \(c\) is 3.

Step 1: Understand the Volume Formula for an Ellipsoid

The volume \( V \) of an ellipsoid with semi-axes lengths \( a \), \( b \), and \( c \) is given by the formula \( V = \frac{4}{3} \pi abc \). In our ellipsoid equation, the semi-axes are \( a = 1 \), \( b = 2 \), and \( c \) which we need to find.

Step 2: Set Up the Equation for the Volume

Substitute the known values into the volume formula. We know the volume is \(8\pi\), so we set up the equation:\[\frac{4}{3} \pi \times 1 \times 2 \times c = 8\pi\] This simplifies to \( \frac{8}{3} \pi c = 8\pi \).

Step 3: Solve for \(c\)

To find \(c\), we first simplify the equation by canceling \(\pi\) on both sides:\[\frac{8}{3} c = 8\] Solve for \(c\) by multiplying both sides by \(\frac{3}{8}\):\[c = 8 \times \frac{3}{8} = 3\] Therefore, \(c = 3\).

Key Concepts

Volume Formula

The volume of an ellipsoid can be determined using a specific formula that involves its semi-axes. An ellipsoid is a three-dimensional shape that looks like a stretched or squashed sphere, defined by its semi-axes: the lengths of its axes. The formula to compute the volume of an ellipsoid is:\[ V = \frac{4}{3} \pi abc \]Where:

• \( a \) is the length of the semi-axis along the x-direction
• \( b \) is the length of the semi-axis along the y-direction
• \( c \) is the length of the semi-axis along the z-direction

In this formula, \( \pi \) is a mathematical constant approximately equal to 3.14159. The factor \( \frac{4}{3} \) is derived from the calculus integration of the volume of a rotated ellipse shape.
To solve problems like determining the unknown semi-axis length or verifying the volume, plug the known values into this formula and solve for the unknowns.

Ellipsoid Semi-Axes

The semi-axes of an ellipsoid play an essential role in its geometric definition and in calculating its volume. Each axis corresponds to one of the three spatial dimensions. In the problem provided, the ellipsoid is defined by the equation:\[ x^{2} + \left( \frac{y}{2} \right)^{2} + \left( \frac{z}{c} \right)^{2} = 1 \]From this equation, we can directly deduce the semi-axes:

• \( a = 1 \)
• \( b = 2 \)
• \( c \) is unknown and needs to be determined

The semi-axes are critical because they provide the unique dimensions of the ellipsoid, influencing its shape and volume.Understanding these semi-axes allows us to manipulate the volume formula effectively to find unknown values. By setting the ellipsoid equation equal to 1, we ensure the proper scaling for each dimension.

Calculus Problem-Solving

Solving for unknowns in an equation involving ellipsoids often requires a calculus-based approach where integration and algebra come together.In particular, manipulating the volume formula for an ellipsoid involves solving a basic algebraic equation once you've substituted the given values into it.For example, in the given problem where the volume of the ellipsoid is specified as \( 8\pi \), we substitute into the formula:\[ \frac{4}{3} \pi \times 1 \times 2 \times c = 8\pi \]Next, by simplifying and solving the equation for \( c \), we cancel \( \pi \) from both sides:\[ \frac{8}{3} c = 8 \]Divide both sides by \( 8/3 \), leading to:\[ c = 8 \times \frac{3}{8} = 3 \]Simple algebraic manipulations like these often transform potentially complex calculus operations into straightforward arithmetic, making it easier to find required values.