Problem 94 - College Algebra with Modeling and Visualization

Question

The volume of a sphere is given by \(V=\frac{4}{3} \pi r^{3},\) where \(r\) is the radius of the sphere. Calculate the volume if the radius is 3 feet. Approximate your answer to the nearest tenth.

Step-by-Step Solution

Answer

The volume is approximately 113.0 cubic feet.

Step 1: Identify the Formula

The formula to calculate the volume of a sphere is given by the expression \( V = \frac{4}{3} \pi r^3 \). Here, \( r \) is the radius of the sphere.

Step 2: Substitute the Radius

We are given that the radius \( r \) of the sphere is 3 feet. Substitute 3 for \( r \) in the formula: \[ V = \frac{4}{3} \pi (3)^3 \]

Step 3: Calculate the Cube of Radius

Calculate \( 3^3 \), which is the cube of the radius:\[ 3^3 = 27 \]

Step 4: Plug into the Formula

Substitute \( 27 \) back into the formula:\[ V = \frac{4}{3} \pi \times 27 \]

Step 5: Simplify the Expression

Simplify the expression by multiplying:\[ V = \frac{4}{3} \times 27 \pi = 36\pi \]

Step 6: Approximate the Volume

Use the approximation \( \pi \approx 3.14 \) to calculate the volume:\[ V \approx 36 \times 3.14 \]

Step 7: Calculate the Final Answer

Perform the multiplication:\[ V \approx 113.04 \] Rounding this to the nearest tenth gives \[ V \approx 113.0 \] cubic feet.

Key Concepts

Sphere

A sphere is a perfectly symmetrical shape in three dimensions. Imagine a ball or a globe. Every point on the surface of a sphere is the same distance from its center. This distance is called the radius. Because of their symmetry, spheres have no edges or corners, and their appearance is consistent from any angle.

Spheres are important in geometry due to their properties. They can be used to model many real-world objects, like planets or bubbles. When learning about the volume of a sphere, it is crucial to understand these properties to comprehend how the formula for volume is derived and applied.

Radius

The radius of a sphere is the distance from the center of the sphere to any point on its surface. It's a key measurement because it helps determine the size of the sphere. In comparison, the diameter is twice the radius and stretches from one side of the sphere to the other, passing through the center.

• The radius provides the crucial linkage between the circle (in two dimensions) and the sphere (in three dimensions).
• To find the volume, you first need to know the radius because it forms the base of calculations in the volume formula.
  

Understanding how to measure or calculate the radius is often the first step in many problems involving spheres.
It simplifies computations and ensures accuracy in applying mathematical models, such as calculating the sphere's volume.

Cubing

Cubing a number means raising it to the power of three. In other words, you multiply the number by itself twice more. Calculating the cube is essential in geometry when determining the volume of a sphere.

When using the volume formula, \( V = \frac{4}{3} \pi r^3 \), the cubing of the radius \( r \) forms the most critical part of the calculation. For instance, if the radius is 3 feet, cubing it results in \( 3^3 = 27 \).

• This computation shows why cubing is used in geometry: to account for three-dimensional space.
• By cubing the radius, you effectively describe how it extends in all directions within the sphere.

Understanding cubing is not just a matter of computation; it's about visualizing how a sphere encompasses space in three dimensions.