The Altitude Of A Right Circular Cylinder Is Twice The Radius Of The Base. Find The Height. If The Volume
Understanding the properties and dimensions of geometric shapes is fundamental in mathematics, especially in the study of three-dimensional figures like cylinders. In this article, we explore a specific problem involving a right circular cylinder where the altitude (height) is twice the radius of its base. Such problems are common in geometry and help reinforce understanding of volume, surface area, and the relationships between different dimensions of cylinders. We will analyze the problem step by step, derive relevant formulas, and demonstrate how to find the height given the volume of the cylinder.
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Understanding the Problem Statement
Restating the Given Data
The problem states:
- The altitude (height) of a right circular cylinder is twice the radius of its base.
- The volume of the cylinder is given (though the specific volume value is not provided in the initial statement — we will assume it is given or work through the general solution).
From this, we can extract the key variables:
- Let r be the radius of the base of the cylinder.
- Since the altitude is twice the radius, h = 2r.
- The volume V is expressed in terms of r (and h).
Our goal is to find the height h when the volume is known, or to express h in terms of volume and radius.
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Formulating the Volume of a Cylinder
Basic Volume Formula
The volume V of a right circular cylinder is given by:
\[ V = \pi r^2 h \]
where:
- \( r \) = radius of the base,
- \( h \) = height of the cylinder,
- \( \pi \) ≈ 3.1416.
Given the problem's condition that h = 2r, we can substitute this into the volume formula:
\[ V = \pi r^2 (2r) = 2 \pi r^3 \]
This simplifies the problem to relating volume directly to the radius:
\[ V = 2 \pi r^3 \]
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Expressing the Height in Terms of Volume and Radius
Solving for the Radius
Suppose the volume V is known; then:
\[ V = 2 \pi r^3 \]
Rearranged to solve for r:
\[ r^3 = \frac{V}{2 \pi} \]
\[
r = \sqrt[3]{\frac{V}{2 \pi}}
\]
Once the radius is found, the height h can be computed using:
\[ h = 2r \]
Expressing the Height in Terms of Volume
Substituting for r:
\[ h = 2 \times \sqrt[3]{\frac{V}{2 \pi}} \]
This formula allows us to find the height directly once the volume V is known.
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Worked Example: Calculating Height for a Given Volume
Let's consider an example where the volume of the cylinder is V = 1000 cubic units.
Step 1: Find the Radius \( r \)
\[
r = \sqrt[3]{\frac{1000}{2 \pi}} = \sqrt[3]{\frac{1000}{2 \times 3.1416}} = \sqrt[3]{\frac{1000}{6.2832}}
\]
Calculating:
\[
\frac{1000}{6.2832} \approx 159.15
\]
Then:
\[
r \approx \sqrt[3]{159.15} \approx 5.36
\]
Step 2: Find the Height \( h \)
\[
h = 2r \approx 2 \times 5.36 \approx 10.72
\]
Answer: The height of the cylinder is approximately 10.72 units when the volume is 1000 cubic units.
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General Approach and Key Points
Summary of Steps to Solve Such Problems
- Identify the given data: volume \( V \), relationship between height and radius \( h = 2r \).
- Write the volume formula: \( V = \pi r^2 h \).
- Substitute \( h = 2r \) into the volume formula to get \( V = 2 \pi r^3 \).
- Solve for \( r \): \( r = \sqrt[3]{\frac{V}{2 \pi}} \).
- Calculate \( r \) and then find \( h = 2r \).
Important Considerations
- The volume must be positive; hence, \( V > 0 \).
- The radius \( r \) derived must be real and positive for the dimensions to make sense physically.
- If the volume is specified, the problem reduces to computing the cube root, which may involve approximation.
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Additional Notes on the Geometry of the Cylinder
Surface Area of the Cylinder
While the problem focuses on volume and height, understanding surface area is also useful:
\[ \text{Surface Area} = 2 \pi r^2 + 2 \pi r h \]
Given \( h = 2r \):
\[ \text{Surface Area} = 2 \pi r^2 + 2 \pi r (2r) = 2 \pi r^2 + 4 \pi r^2 = 6 \pi r^2 \]
This reveals that for this specific relationship between height and radius, the surface area depends solely on the radius.
Implications of the Relationship \( h = 2r \)
- The height is directly proportional to the radius.
- Increasing the radius \( r \) scales both the volume and surface area accordingly.
- The problem exemplifies how constraints between dimensions simplify calculations.
Conclusion
In summary, when the altitude of a right circular cylinder is twice the radius of its base, the problem of finding the height given the volume becomes straightforward through algebraic substitution. The key steps involve expressing the volume formula in terms of a single variable, solving for the radius, and then deriving the height. The general formula for the height in terms of volume is:
\[ h = 2 \times \sqrt[3]{\frac{V}{2 \pi}} \]
This approach not only provides a clear pathway to solving such problems but also deepens understanding of the relationships between the dimensions of cylinders. Whether in academic exercises or real-world applications like engineering and design, mastering these relationships is essential for accurate measurements and calculations.
