The Volume Of The Cylinder If 500pi Cubic Inches And The Radius Is 5 Inches What Is The Height Of The
Understanding the relationship between the volume, radius, and height of a cylinder is fundamental in geometry and practical applications such as engineering, manufacturing, and architecture. In this article, we will explore how to determine the height of a cylinder when given its volume and radius, using the formula for the volume of a cylinder. Specifically, we will analyze the problem: The volume of the cylinder is 500π cubic inches, and the radius is 5 inches. What is the height of the cylinder?
By dissecting this problem, we will cover the essential concepts, formula derivation, step-by-step calculations, and additional considerations to deepen your understanding of cylindrical volume calculations.
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Understanding the Cylinder Volume Formula
The Basic Formula
The volume \(V\) of a cylinder is calculated using the formula:
\[
V = \pi r^2 h
\]
Where:
- \(V\) is the volume of the cylinder,
- \(r\) is the radius of the base,
- \(h\) is the height of the cylinder,
- \(\pi\) is the mathematical constant Pi, approximately 3.14159.
This formula states that the volume is proportional to the area of the circular base (\(\pi r^2\)) times the height.
Implications of the Formula
- The volume increases linearly with the height; doubling the height doubles the volume.
- The volume is proportional to the square of the radius; increasing the radius significantly increases the volume.
- When the volume and radius are known, the height can be isolated and calculated directly.
Given Data and Goal
Problem Restatement
- The volume \(V = 500\pi\) cubic inches.
- The radius \(r = 5\) inches.
- The unknown: height \(h\).
Objective
Calculate the height \(h\) of the cylinder using the provided data.
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Step-by-Step Calculation of the Cylinder Height
Step 1: Write Down the Volume Formula
\[
V = \pi r^2 h
\]
Substitute known values:
\[
500\pi = \pi \times (5)^2 \times h
\]
Step 2: Simplify the Equation
Calculate \(r^2\):
\[
(5)^2 = 25
\]
So,
\[
500\pi = \pi \times 25 \times h
\]
Step 3: Isolate \(h\)
Divide both sides of the equation by \(\pi \times 25\):
\[
h = \frac{500\pi}{\pi \times 25}
\]
Since \(\pi\) appears in numerator and denominator, they cancel out:
\[
h = \frac{500}{25}
\]
Step 4: Final Calculation
\[
h = 20
\]
Thus, the height of the cylinder is 20 inches.
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Understanding the Result and Its Significance
Final Answer
The height of the cylinder is 20 inches.
Interpretation
- Given the volume and radius, the height can be easily computed using the volume formula.
- The result indicates that a cylinder with a radius of 5 inches and volume \(500\pi\) cubic inches must be 20 inches tall.
Practical Implications
- This calculation is applicable in designing cylindrical containers, tanks, or pipes where volume specifications are critical.
- Knowing how to manipulate the volume formula allows engineers and designers to customize dimensions according to volume constraints.
Additional Considerations and Variations
Different Volume Units or Dimensions
- The same principles apply if the volume is given in different units, provided all measurements are consistent.
- For example, if the volume was in liters and the radius in centimeters, conversions would be necessary.
Handling Different Radii or Volumes
- When the radius varies, the same formula applies, and solving for height involves rearranging the formula accordingly.
- For example, if the volume and height are known, the radius can be calculated.
Example: Calculating the Radius Given Volume and Height
Suppose the volume is known, and height is given; then:
\[
r = \sqrt{\frac{V}{\pi h}}
\]
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Summary and Key Takeaways
- The volume of a cylinder is directly proportional to the square of its radius and its height.
- When given the volume and radius, the height can be straightforwardly calculated by rearranging the volume formula.
- In the specific problem, with a volume of \(500\pi\) cubic inches and a radius of 5 inches, the height is 20 inches.
- Mastery of the volume formula allows for solving a wide array of practical problems involving cylindrical objects.
Conclusion
Understanding how to determine the height of a cylinder from its volume and radius is a fundamental skill in geometry that finds relevance in numerous real-world applications. By applying the formula \(V = \pi r^2 h\), substituting known values, and performing algebraic manipulations, one can solve for any unknown dimension of a cylinder efficiently. The specific problem presented demonstrates this process clearly, yielding a concise and practical result: the height of the cylinder is 20 inches. This knowledge not only enhances mathematical proficiency but also empowers practical decision-making in fields that require precise volumetric measurements of cylindrical objects.
