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Chapter 8: Problem 21

find the volume of each cone. Round to the nearest tenth. \(r=3\) in. \(, h=4\) in

Short Answer

Expert verified
The volume is approximately \(37.7\) cubic inches.

Step by step solution

01

Write down the formula for the volume of a cone

The formula to find the volume of a cone is: \[ V = \frac{1}{3} \pi r^2 h \]
02

Substitute the given values into the formula

Put the given radius (\(r=3\) inches) and height (\(h=4\) inches) into the cone volume formula: \[ V = \frac{1}{3} \pi (3)^2 (4) \]
03

Calculate the expression inside the formula

First, calculate the square of the radius: \[ 3^2 = 9 \] Then multiply by the height: \[ 9 \times 4 = 36 \] So, the expression becomes: \[ V = \frac{1}{3} \pi \times 36 \]
04

Final calculation

Multiply 36 by \( \frac{1}{3} \): \[ \frac{1}{3} \times 36 = 12 \] So, the volume is: \[ V = 12 \pi \] Finally, round to the nearest tenth: \[ V \approx 37.7 \, \text{in}^3 \] using \(\pi \approx 3.14159\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

geometry
Geometry is a branch of mathematics that deals with the properties and relationships of points, lines, surfaces, and solids. In the case of the cone volume problem, we are working with a 3-dimensional shape called a cone. Cones have a circular base and a single apex or vertex. Understanding the properties of a cone helps us calculate its volume.
cone volume formula
The volume of a cone can be found using a specific formula. This formula is rooted in geometry and helps us calculate how much space is inside the cone. The formula for the volume of a cone is: \( V = \frac{1}{3} \pi r^2 h \). In this formula, \( V \) represents the volume, \( r \) is the radius of the circular base, and \( h \) is the height of the cone. To use this formula, we substitute the given values for the radius and the height into the formula.
mathematical calculations
To find the volume of the cone, we perform a series of mathematical calculations. Let's walk through them step-by-step:
1. First, we square the radius. For a radius of 3 inches, this looks like: \( 3^2 = 9\).
2. Next, we multiply this result by the height of the cone; here, the height is 4 inches, so: \( 9 \times 4 = 36 \).
3. Now, we need to multiply this product by \( \pi \) (approximately 3.14159), and then take one-third of this value. This results in: \( \frac{1}{3} \pi \times 36 = 12 \pi \).
4. Finally, rounding 12蟺 to the nearest tenth, we get \( V \approx 37.7 \) cubic inches.

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