Problem 69 - Prealgebra

Question

The radius of a sphere is 3.9 inches. Find the volume to the nearest hundredth.

Step-by-Step Solution

Answer

The volume of the sphere is approximately 248.43 cubic inches.

Step 1: Understand the Volume Formula

The formula for the volume of a sphere is given by \( V = \frac{4}{3} \pi r^3 \), where \( r \) is the radius of the sphere. To find the volume, we will substitute the given radius into this formula.

Step 2: Substitute the Radius

Given that the radius \( r = 3.9 \) inches, substitute this value into the volume formula: \[ V = \frac{4}{3} \pi (3.9)^3 \]

Step 3: Calculate \( r^3 \)

Calculate \( (3.9)^3 \). This means multiplying 3.9 by itself three times: \( 3.9 \times 3.9 \times 3.9 = 59.319 \).

Step 4: Calculate \( \frac{4}{3} \pi r^3 \)

Substitute \( r^3 = 59.319 \) into the formula and simplify: \[ V = \frac{4}{3} \pi (59.319) \] Using \( \pi \approx 3.14 \), calculate:\[ V \approx \frac{4}{3} \times 3.14 \times 59.319 \] \[ V \approx 248.432 \] cubic inches.

Step 5: Round the Result

Round the calculated volume to the nearest hundredth. The volume calculated was approximately 248.432. Rounded to the nearest hundredth, it is 248.43 cubic inches.

Key Concepts

Understanding Radius Calculation

In geometry, a sphere is a perfectly round 3D shape where every point on its surface is the same distance from its center. This distance is what we call the radius. Calculating the radius effectively is crucial since it often forms the basis for more advanced geometry calculations, such as finding the volume or surface area of the shape.

Remember:

• The radius is the distance from the center of the sphere to any point on its surface.
• Knowing the radius allows you to use specific mathematical formulas, like the volume of a sphere formula, confidently.
• The formula to find the volume of a sphere, as seen, is dependent on knowing the radius: \( V = \frac{4}{3} \pi r^3 \).

While it may seem like a small detail, knowing how to measure and apply the radius is fundamental to tackling many geometry problems involving spheres.

Solving Geometry Problems with Spheres

Geometry problems can often seem tricky, but breaking them down into steps helps simplify them. When dealing with spheres, understanding the basic attributes such as the radius makes problems more approachable.

Here's how to approach geometry problems involving spheres:

• Identify what measurements you have (like the radius) and what you need to find (like volume or surface area).
• Use the known measurements to substitute into established formulas. Substitution helps turn abstract formulas into concrete numbers.
• Ensure to solve calculation steps one at a time, as this reduces errors and builds accuracy.

When faced with a sphere's problem, remember to stay calm and follow patterns. Each step in a problem-solving process builds on the one before it, like deciphering a puzzle.

Utilizing Mathematical Formulas

Mathematical formulas are tools that provide the framework to solve geometry problems efficiently. For spheres, the key formula to remember is the volume formula, which utilizes the radius:
\[V = \frac{4}{3} \pi r^3\]Where:

• \(V\) is the volume of the sphere.
• \(\pi\) (Pi) is a constant approximately equal to 3.14.
• \(r\) is the radius of the sphere.

Formulas turn complex concepts into straightforward calculations. Each element in a formula is vital, and correctly substituting known values is key to solving any geometric computation. By consistently practicing these formulas, solving problems becomes a quicker and more intuitive task. Remember, understanding the formula is as crucial as the calculation itself, as it allows for flexibility and adaptation when encountering new or similar problems in the future.