Question

What is the volume of a hemisphere with a radius of \(4 \mathrm{ft}\) (take \(\pi\) to be \(3.14\) and work to the nearest full \(\mathrm{ft}^3\) )?

Short answer

The volume of the hemisphere is 134 ft^3.

Step-by-step solution

Understand the Formula for the Hemisphere Volume

The volume of a hemisphere is given by the formula: \[ V = \frac{2}{3} \pi r^3 \] where \( r \) is the radius of the hemisphere.

Substitute the Radius into the Formula

Given that the radius (\( r \)) is 4 feet: \[ V = \frac{2}{3} \pi (4)^3 \]

Calculate the Cube of the Radius

First, calculate \( 4^3 \): \[ 4^3 = 4 \times 4 \times 4 = 64 \]

Substitute the Value back into the Formula

Now, substitute 64 back into the volume formula: \[ V = \frac{2}{3} \pi \times 64 \]

Simplify the Expression

Next, simplify the expression: \[ V = \frac{2}{3} \times 3.14 \times 64 \]

Perform the Multiplication

Calculate the multiplication step by step: \[ \frac{2}{3} \times 3.14 = 2.0933 \] and then: \[ 2.0933 \times 64 = 133.96 \] ft^3.

Round to the Nearest Full Cubic Foot

Round 133.96 ft^3 to the nearest whole number: \[ 134 \; \text{ft}^3 \]

Key concepts

mathematics problems

Understanding and solving mathematics problems often requires breaking them down into smaller, manageable steps. This is especially true for geometry and volume calculations. The key to successfully solving these problems is a solid grasp of the relevant formulas and concepts. In this exercise, the goal was to calculate the volume of a hemisphere with a given radius of 4 feet. By methodically applying the correct formula and performing precise calculations, we arrived at the solution. Remember, practice makes perfect. The more you work on similar problems, the better you'll get at quickly navigating through the steps required to find the solution.

geometry

Geometry is the branch of mathematics concerned with the properties and relations of points, lines, surfaces, and solids. When dealing with volumes, we often work with three-dimensional shapes such as spheres, cubes, and cylinders. A hemisphere is essentially half of a sphere. To understand its volume, you need to comprehend the properties of spheres and how volumes are calculated for these shapes. In this exercise, we relied on the formula that relates the radius of a hemisphere to its volume. Mastering these geometric principles is crucial for solving real-world problems involving space and measurement. Break down complex terms and visualize the shape to improve your grasp of geometry.

volume formulas

Volume formulas are essential tools in geometry for calculating the space occupied by three-dimensional objects. The volume of a hemisphere can be calculated using the formula \[ V = \frac{2}{3} \pi r^3 \] where \( r \) is the radius of the hemisphere. This formula is derived from the volume of a sphere, which is given by \[ V_{sphere} = \frac{4}{3} \pi r^3 \], Since a hemisphere is half of a sphere, its volume is half that of the sphere. Proper application of these formulas and careful substitution of the given values are key to solving such problems. In our exercise, substituting 4 feet for \(r\) and simplifying the expression step by step led us to the final volume: 134 cubic feet. Familiarize yourself with these formulas and practice regularly.