Problem 35 - Thinking Mathematically

Question

Find the ratio, reduced to lowest terms, of the volume of a sphere with a radius of 3 inches to the volume of a sphere with a radius of 6 inches.

Step-by-Step Solution

Answer

The ratio of the volume of a sphere with a radius of 3 inches to the volume of a sphere with a radius of 6 inches, in simplest form, is 1:8.

Step 1: Compute the Volume of the First Sphere

Using the volume formula of a sphere,e plug in the radius of the first sphere, which is 3 inches. This gives \( V_1 = \frac{4}{3} \pi (3)^3 \). Therefore, \( V_1 = 36\pi \) cubic inches.

Step 2: Compute the Volume of the Second Sphere

Similarly, plug in the radius of the second sphere, which is 6 inches, into the volume formula. This gives \( V_2 = \frac{4}{3} \pi (6)^3 \). Therefore, \( V_2 = 288\pi \) cubic inches.

Step 3: Form the Ratio

Now, form the ratio of the two volumes. This is \( \frac{V_1}{V_2} = \frac{36\pi}{288\pi} \). The \( \pi \)s cancel out during the division.

Step 4: Simplify the Ratio

The ratio is simplified to lowest terms by dividing both numbers by their greatest common divisor. This gives \( \frac{36}{288} = \frac{1}{8} \). So, the ratio of the two volumes in simplest form is 1:8.

Key Concepts

Volume of a Sphere

The volume of a sphere is a measure of the space occupied by a spherical object. To calculate the volume of a sphere, you use a specific mathematical formula. The formula for finding the volume of a sphere is \( V = \frac{4}{3} \pi r^3 \), where \( V \) is the volume and \( r \) is the radius of the sphere.

This formula requires knowing only one value—the radius. The radius is the distance from the center of the sphere to any point on its surface. When you plug in the value of the radius into the formula, the math works like magic to give you the volume!

Remember that \( \pi \approx 3.14159 \), but often we just leave it as \( \pi \) in expressions because it simplifies calculations when you compare or reduce ratios.

Ratio

A ratio is a way to compare two quantities by expressing how many times one quantity is contained within another. In our exercise, we compared the volumes of two spheres. Ratios are written in the form of two numbers separated by a colon, like this: 1:8.

To find the ratio in simplest form, you compare the two volumes obtained from the volume formula. By dividing the first volume by the second volume, you create a fraction. Simplifying this fraction helps in finding the simplest form of the ratio.

Ratios are very helpful in making comparisons more insightful since they reduce complicated quantities into manageable comparisons.

Simplifying Fractions

Simplifying fractions means breaking down a fraction to its simplest form, where no further division by common factors is possible. In the exercise, we simplified the ratio obtained from the volumes, \( \frac{36}{288} \), by finding the greatest common divisor (GCD) of the numerator and the denominator.

To simplify, divide both the numerator and the denominator by their GCD. For the numbers 36 and 288, this was 36, resulting in the fraction \( \frac{1}{8} \).

Simplifying fractions is very useful in mathematical problem solving because it makes numbers easier to work with and comparisons clearer.

Mathematical Problem Solving

Mathematical problem solving is the process of finding solutions to complex problems using logical reasoning and mathematical techniques. The exercise used here is an example of a problem solved by applying mathematical concepts like volume calculation, ratio determination, and fraction simplification.

The key steps in solving such problems involve understanding the question, applying relevant formulas and techniques, simplifying complex expressions, and finally, interpreting the results.

Developing a step-by-step approach is crucial, as it helps break down the problem into manageable parts. This, in turn, ensures each step is correctly executed, leading to the accurate solution—like finding the simplest form of a ratio in the given exercise.