Question

Sales The Boy Scouts of America recently made a giant popcorn ball to promote their popcorn sales. Scouts and other helpers added to the popcorn ball as it traveled around the country. a. What was the volume of the popcorn ball when it had a diameter of 6 feet? Round to the nearest hundredth. b. Suppose the diameter of the popcorn ball was enlarged to 9 feet. What would be its volume to the nearest hundredth? c. What was the volume of the popcorn needed to increase the popcorn ball from a diameter of 6 feet to a diameter of 9 feet?

Short answer

a: 113.10 cubic feet, b: 381.70 cubic feet, c: 268.60 cubic feet.

Step-by-step solution

Understand the formula for volume of a sphere

The volume of a sphere is calculated using the formula \( V = \frac{4}{3} \pi r^3 \), where \( r \) is the radius of the sphere.

Calculate the radius from the diameter

The diameter is given, so we calculate the radius by dividing the diameter by 2. For part (a), the diameter is 6 feet, so \( r = \frac{6}{2} = 3 \) feet. For part (b), the diameter is 9 feet, so \( r = \frac{9}{2} = 4.5 \) feet.

Calculate the volume for a diameter of 6 feet

Using the formula \( V = \frac{4}{3} \pi r^3 \), substitute \( r = 3 \). This gives: \( V = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi \, 27 = 36\pi \approx 113.10 \) cubic feet.

Calculate the volume for a diameter of 9 feet

Substitute \( r = 4.5 \) into the volume formula: \( V = \frac{4}{3} \pi (4.5)^3 = \frac{4}{3} \pi \, 91.125 \approx 381.70 \) cubic feet.

Calculate the difference in volume

To find the volume of popcorn needed to increase from a diameter of 6 feet to 9 feet, subtract the volume at 6 feet from the volume at 9 feet: \( 381.70 - 113.10 \approx 268.60 \) cubic feet.

Key concepts

Volume of a Sphere

The volume of a sphere is a crucial concept in geometry and is essential when dealing with spherical objects. The formula for finding the volume of a sphere is given by: \[ V = \frac{4}{3} \pi r^3 \]Here, \( V \) represents the volume, and \( r \) is the radius of the sphere. It's important to understand that this formula helps in determining how much space a sphere occupies, a concept referred to as volume. This concept is applied in various real-world scenarios, such as determining the amount of air in a balloon, or, in this case, the popcorn in a giant popcorn ball. Calculating the volume of a sphere accurately is key to many projects, especially those involving packaging, manufacturing, or sales promotions.

Mathematical Formulas

Mathematical formulas are the backbone of problem-solving in geometry. They provide a structured way to approach and solve problems. In this scenario, the formula for volume of a sphere, \( V = \frac{4}{3} \pi r^3 \), is utilized to find the answer to a problem related to spheres. Here’s how it works:

• Identify what the formula requires: the radius of the sphere.
• Substitute the known values, like the radius, into the formula.
• Perform the arithmetic operations carefully, respecting the order of operations.

The beauty of formulas is their consistency and reliability. Knowing the mathematical formula means you can tackle similar problems with confidence. It allows you to convert a physical scenario into a mathematical one, enabling precise calculations.

Diameter and Radius

Understanding the concepts of diameter and radius is fundamental in geometry, especially when dealing with circular shapes like spheres. The diameter is the straight-line distance passing through the center of a circle or sphere, connecting opposite sides. The radius, however, is half of the diameter, running from the center to any point on the edge.

Calculating Radius from Diameter

To find the radius when you know the diameter, simply divide the diameter by 2. For instance, if a sphere has a diameter of 6 feet, the radius would be \( \frac{6}{2} \) feet, which equals 3 feet. This relationship is vital because most formulas, including the volume of a sphere, require you to input the radius, not the diameter. Thus, converting the diameter to a radius is often the first step in solving related problems.

Calculating Volume Differences

When dealing with changes in size, calculating the differences in volume is a practical application of understanding a sphere's volume. In this case, the problem involves finding out how much more popcorn is needed when the diameter of a popcorn ball increases.

Steps to Calculate Volume Differences

First, calculate the volume for each size using the volume formula with their respective radii.

Then, subtract the smaller volume from the larger volume.

This gives you the increase in volume, i.e., the amount of additional material needed.

Understanding volume differences is crucial in tasks like this because it gives insight into how a small change in dimensions can significantly alter the total volume. In practical situations, this means estimating costs, storage needs, and resources effectively.