Question

Ahmed is creating a large balloon in the shape of a medical capsule for a drug company's Employee Day festivities. The balloon is a cylinder 50 feet long and 12 feet in diameter, capped at each end by half spheres 12 feet in diameter. How many cubic feet of gas is needed to fill the balloon? A. 6557 cubic feet B. 9623 cubic feet C. 13,137 cubic feet D. 22,405 cubic feet

Short answer

A. 6557 cubic feet

Step-by-step solution

Understand the Problem

The balloon is composed of a cylinder and two half spheres. The total volume of the balloon will be the sum of the volume of the cylinder and the volume of the two half spheres.

Find the Volume of the Cylinder

The formula for the volume of a cylinder is \( V_{\text{cylinder}} = \pi r^2 h \). The radius (r) is half the diameter, which is \(12/2 = 6 \) feet. The height (h) is 50 feet. Therefore, \[ V_{\text{cylinder}} = \pi (6)^2 (50) = 1800\pi \].

Find the Volume of One Sphere

The formula for the volume of a sphere is \( V_{\text{sphere}} = \frac{4}{3}\pi r^3 \). The radius (r) is 6 feet. Therefore, \[ V_{\text{sphere}} = \frac{4}{3}\pi (6)^3 = 288\pi \].

Find the Volume of Both Half Spheres

Since each end is a half sphere, two half spheres make one full sphere. So, the volume of both half spheres combined is \( 288\pi \).

Calculate the Total Volume

Add the volume of the cylinder and the volume of the sphere: \[ V_{\text{total}} = 1800\pi + 288\pi = 2088\pi \]. Since \( \pi \approx 3.14159 \), we get \[ V_{\text{total}} \approx 2088 \times 3.14159 = 6556.9 \text{ cubic feet} \].

Determine the Closest Answer

The closest answer to 6557 cubic feet is A. 6557 cubic feet.

Key concepts

Geometry

Geometry is a branch of mathematics that deals with shapes, sizes, and the properties of space. It's all about understanding how figures, like circles and triangles, and solids, like cylinders and spheres, interact and fit together. In this exercise, we deal with a combination of cylindrical and spherical shapes to find the volume of a balloon.

**Key Concepts in Geometry**
Understanding geometry involves

• Identifying shapes
• Calculating areas and volumes
• Understanding properties like parallelism and perpendicularity

In particular, our focus here is on finding volumes, which is a measure of the space a 3-dimensional object occupies.

Cylinder Volume

Finding the volume of a cylinder involves a straightforward formula: \( V_{\text{cylinder}} = \pi r^2 h \). Let's break this down step-by-step.

**Components of the Formula**

• \( \pi \): A constant approximately equal to 3.14159
• \( r \): The radius of the cylinder's base. For this exercise, the radius is half the diameter, so it’s 6 feet.
• \( h \): The height of the cylinder. Here, it's 50 feet.

So, for Ahmed's balloon, substituting these values gives: \ V_{\text{cylinder}} = \pi (6)^2 (50) = 1800\pi \
The unit is cubic feet because we are dealing with three-dimensional space.

Sphere Volume

When calculating the volume of a sphere, the formula is a bit different: \ V_{\text{sphere}} = \frac{4}{3}\pi r^3 \.

**Components of the Formula**

• \( \pi \): The same constant as in the cylinder formula
• \( r \): The radius of the sphere, which is also 6 feet in this exercise

By substituting the radius, we get: \ V_{\text{sphere}} = \frac{4}{3}\pi (6)^3 = 288\pi \

But keep in mind that Ahmed's balloon consists of two half-spheres (one at each end). Adding these two halves gives us the volume of one full sphere: \ 288\pi \ cubic feet.

GED Math

The General Educational Development (GED) math test covers many essential math skills, including geometry. Understanding how to apply formulas for volume is a crucial part of this.

**Important Skills for the GED Math Test**

• Interpreting and solving word problems
• Applying mathematical formulas
• Performing accurate calculations

In this example, Ahmed's balloon problem helps practice these skills:

• Reading the problem carefully to identify the shapes involved
• Using geometry formulas accurately
• Summing the volumes of different shapes

These steps are essential for success not only in this specific problem but also in many real-world scenarios you may encounter.