Problem 48 - Elementary Algebra

Question

Applications involving variation. The volume of a sphere varies directly as the cube of its radius. When the radius of a sphere measures 3 meters, the volume is \(36 \pi\) cubic meters. Find the volume of a sphere with radius 1 meter.

Step-by-Step Solution

Answer

The volume is \( \frac{4 \pi}{3} \) cubic meters.

Step 1: State the Direct Variation Formula

When the volume of a sphere varies directly as the cube of its radius, it means we can express this relationship as: \[ V = k r^3 \] where \( V \) is the volume, \( r \) is the radius, and \( k \) is a constant of variation.

Step 2: Use Given Values to Find the Constant (k)

We are given that when the radius is 3 meters, the volume is \( 36 \pi \) cubic meters. Substitute these into the formula to find \( k \): \[ 36 \pi = k (3)^3 \] \[ 36 \pi = k imes 27 \] Solve for \( k \): \[ k = \frac{36 \pi}{27} = \frac{4 \pi}{3} \]

Step 3: Calculate the Volume for Radius 1 Meter

With \( k = \frac{4 \pi}{3} \), find the volume when the radius is 1 meter:\[ V = k (1)^3 \] \[ V = \frac{4 \pi}{3} \times 1 = \frac{4 \pi}{3} \]

Step 4: Present the Result

Thus, the volume of a sphere with radius 1 meter is \( \frac{4 \pi}{3} \) cubic meters.

Key Concepts

Volume of a Sphere

The volume of a sphere is an important concept in geometry that describes the amount of space contained within a spherical object. Understanding this concept can help you solve various real-world problems related to spheres, like calculating how much air can fill a basketball, or how much water is needed to fill a spherical tank.

To calculate the volume of a sphere, you need to use the formula:

• \( V = \frac{4}{3} \pi r^3 \)

This equation shows that the volume \( V \) depends on the radius \( r \) of the sphere and the mathematical constant \( \pi \).

It's the radius of the sphere raised to the power of three, which underscores the three-dimensional nature of the sphere. The term \( \frac{4}{3} \) is a particular mathematical fraction that ensures the volume is correctly adjusted for the spherical shape, unlike a cube or other geometrical figures.

Constant of Variation

When we say that the volume of a sphere varies directly with the cube of its radius, we introduce the notion of a constant of variation, \( k \). This constant represents how the volume scales with changes in the radius, and it remains unchanged for similar conditions among spheres.

In our context here, the formula expressing this relationship is:

• \( V = k r^3 \)

You can think of \( k \) as a fixed factor that modifies how the radius is converted into volume. Finding \( k \) requires given values, like when we know the volume and radius of a specific sphere. By substituting these known values into the equation, we can solve for \( k \). For example, with the given radius 3 meters, and a volume of \( 36 \pi \) cubic meters, we calculated \( k \) as \( \frac{4 \pi}{3} \).

This calculated constant then helps us find the volume of other spheres with different radii, assuming they follow the same direct variation pattern.

Radius and Volume Relationship

The relationship between the radius and the volume of a sphere is one of direct variation to the cube of the radius. This means that if you double the radius of a sphere, its volume will increase eightfold, because you raise the radius to the power of three.

Understanding this relationship is crucial when dealing with spheres, as any change in the radius will significantly affect the volume. For instance:

• If \( r = 1 \), \( V = \frac{4 \pi}{3} \)
• If \( r = 2 \), \( V = \frac{4 \pi}{3} \times 8 = \frac{32 \pi}{3} \)
• If \( r = 3 \), \( V = \frac{4 \pi}{3} \times 27 = 36 \pi \)

This relationship showcases the exponential increase in volume with changes in the radius, demonstrating why knowing the radius is fundamental when calculating a sphere's volume. By visually understanding how radius variations affect volume, it is easier to conceptualize the resulting changes and ensure accurate calculations.