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Magazine Chapter 12: Problem 27
Find the volume of each cone described. Round to the nearest tenth. (lesson \(11-3\) ) radius \(7 \mathrm{cm},\) height \(9 \mathrm{cm}\)
Short Answer
Expert verified
The volume of the cone is approximately 461.8 cm鲁.
Step by step solution
01
Understanding the Cone's Volume Formula
The volume of a cone is calculated using the formula \( V = \frac{1}{3} \pi r^2 h \), where \( r \) is the radius of the base, and \( h \) is the height of the cone. Our goal is to apply this formula to find the volume.
02
Substitute Values into the Volume Formula
Given the radius \( r = 7 \) cm and height \( h = 9 \) cm, substitute these values into the formula.\[ V = \frac{1}{3} \pi (7)^2 (9) \]
03
Calculate the Area of the Base
First, calculate the area of the base \( \pi r^2 \), which is part of the volume formula.\[ \pi r^2 = \pi \times 7^2 = 49\pi \]
04
Incorporate the Height and Simplify
Multiply the area of the base by the height: \( 49\pi \times 9 \).\[ 49\pi \times 9 = 441\pi \]
05
Apply the Volume Constant
Multiply by \( \frac{1}{3} \, \):\[ V = \frac{1}{3} \times 441\pi = 147\pi \]
06
Calculate the Volume Numerically and Round
Now, compute the numerical value by approximating \( \pi \approx 3.1416 \):\[ V = 147 \times 3.1416 \approx 461.81 \ \text{cm}^3 \]. Round this to the nearest tenth to obtain \( 461.8 \ \text{cm}^3 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Geometry Formulas
Geometry is a vast field with numerous shapes and their respective formulas. In the case of cones, the primary formula that comes into play is the volume, calculated as:\[ V = \frac{1}{3} \pi r^2 h \]Where:
- \( V \) is the volume
- \( r \) is the radius of the base
- \( h \) is the height
Cone Calculations
Calculating the volume of a cone involves clear and systematic steps, just like in our example:
- Start by identifying the radius and height. In our exercise, the radius is 7 cm, and the height is 9 cm.
- Next, compute the area of the base using the formula \( \pi r^2 \), giving us \( 49\pi \).
- Multiply the result by the height (9), leading to \( 441\pi \).
- Finally, multiply the whole expression by \( \frac{1}{3} \) to find the cone's volume: \( 147\pi \).
- By approximating \( \pi \) to 3.1416, the final volume of the cone is roughly 461.81 \( \text{cm}^3 \).
Mathematics Education
Understanding the volume of a cone is a small but significant part of mathematics education. By practicing problems like these, students build essential skills:
- Recognizing and solving geometric problems.
- Applying formulas accurately and confidently.
- Developing analytical thinking through problem-solving.
Measurement
Measurement is foundational to geometry, and it becomes clear in exercises like calculating the volume of a cone. Measuring involves determining dimensions like length, area, and volume, all of which are crucial for understanding space and size.
For cones:
- The radius gives us a sense of the base's size.
- The height determines how tall the cone is.
- Volume combines these measurements to provide total space within.
