Type The Correct Answer In The Box. Use Numerals Instead Of Words.The Surface Area Of A Cone Is $216 \pi$ Square Units. The Height Of The Cone Is $\frac{5}{3}$ Times Greater Than The Radius. What Is The Length Of The Radius Of The

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Introduction

In mathematics, the surface area of a cone is a crucial concept that helps us understand the relationship between the radius, height, and slant height of the cone. Given the surface area of a cone as $216 \pi$ square units and the height being $\frac{5}{3}$ times greater than the radius, we are tasked with finding the length of the radius of the cone. In this article, we will delve into the world of cone geometry and explore the formulas and calculations required to determine the radius of the cone.

Understanding the Surface Area of a Cone

The surface area of a cone is given by the formula:

A=Ï€r2+Ï€rlA = \pi r^2 + \pi r l

where $A$ is the surface area, $r$ is the radius, and $l$ is the slant height of the cone. We are given that the surface area of the cone is $216 \pi$ square units, so we can set up the equation:

216Ï€=Ï€r2+Ï€rl216 \pi = \pi r^2 + \pi r l

Simplifying the Equation

We can simplify the equation by dividing both sides by $\pi$:

216=r2+rl216 = r^2 + r l

Finding the Relationship Between the Radius and Height

We are given that the height of the cone is $\frac{5}{3}$ times greater than the radius. Let's denote the radius as $r$ and the height as $h$. Then, we can write:

h=53rh = \frac{5}{3} r

Substituting the Relationship into the Equation

We can substitute the relationship between the height and radius into the simplified equation:

216=r2+r(53r)216 = r^2 + r \left(\frac{5}{3} r\right)

Simplifying the Equation Further

We can simplify the equation further by multiplying the terms:

216=r2+53r2216 = r^2 + \frac{5}{3} r^2

Combining Like Terms

We can combine the like terms:

216=83r2216 = \frac{8}{3} r^2

Solving for the Radius

We can solve for the radius by dividing both sides by $\frac{8}{3}$:

r2=21683r^2 = \frac{216}{\frac{8}{3}}

Simplifying the Expression

We can simplify the expression by multiplying the numerator and denominator by $3$:

r2=6488r^2 = \frac{648}{8}

Simplifying Further

We can simplify the expression further by dividing the numerator and denominator by $8$:

r2=81r^2 = 81

Taking the Square Root

We can take the square root of both sides to find the radius:

r=81r = \sqrt{81}

Simplifying the Expression

We can simplify the expression by evaluating the square root:

r=9r = 9

Conclusion

In this article, we have calculated the radius of a cone given its surface area and the relationship between the height and radius. We have used the formulas and calculations to determine the radius of the cone, and we have found that the length of the radius is $9$ units. This problem requires a deep understanding of cone geometry and the ability to apply mathematical formulas to solve real-world problems.

Discussion

This problem is a great example of how mathematics can be applied to real-world problems. The surface area of a cone is an important concept in engineering and architecture, and being able to calculate it is crucial for designing and building structures. The relationship between the height and radius of a cone is also an important concept in mathematics, and it is used in many different areas of study.

References

  • [1] "Geometry of Cones" by Math Open Reference
  • [2] "Surface Area of a Cone" by Math Is Fun
  • [3] "Cones and Spheres" by Khan Academy

Additional Resources

  • [1] "Cones and Cylinders" by IXL
  • [2] "Surface Area and Volume of Cones" by Purplemath
  • [3] "Geometry of Cones" by Wolfram Alpha
    Q&A: Calculating the Radius of a Cone =====================================

Introduction

In our previous article, we calculated the radius of a cone given its surface area and the relationship between the height and radius. In this article, we will answer some frequently asked questions related to calculating the radius of a cone.

Q: What is the formula for the surface area of a cone?

A: The formula for the surface area of a cone is:

A=Ï€r2+Ï€rlA = \pi r^2 + \pi r l

where $A$ is the surface area, $r$ is the radius, and $l$ is the slant height of the cone.

Q: How do I find the slant height of a cone?

A: To find the slant height of a cone, you can use the Pythagorean theorem:

l=r2+h2l = \sqrt{r^2 + h^2}

where $l$ is the slant height, $r$ is the radius, and $h$ is the height of the cone.

Q: What is the relationship between the height and radius of a cone?

A: The relationship between the height and radius of a cone is given by:

h=53rh = \frac{5}{3} r

Q: How do I calculate the radius of a cone given its surface area and the relationship between the height and radius?

A: To calculate the radius of a cone given its surface area and the relationship between the height and radius, you can use the following steps:

  1. Set up the equation:

216=r2+rl216 = r^2 + r l

  1. Substitute the relationship between the height and radius:

216=r2+r(53r)216 = r^2 + r \left(\frac{5}{3} r\right)

  1. Simplify the equation:

216=r2+53r2216 = r^2 + \frac{5}{3} r^2

  1. Combine like terms:

216=83r2216 = \frac{8}{3} r^2

  1. Solve for the radius:

r2=21683r^2 = \frac{216}{\frac{8}{3}}

  1. Simplify the expression:

r2=81r^2 = 81

  1. Take the square root:

r=81r = \sqrt{81}

  1. Simplify the expression:

r=9r = 9

Q: What are some real-world applications of calculating the radius of a cone?

A: Calculating the radius of a cone has many real-world applications, including:

  • Designing and building structures such as bridges, buildings, and towers
  • Calculating the volume of a cone
  • Determining the surface area of a cone
  • Understanding the relationship between the height and radius of a cone

Q: What are some common mistakes to avoid when calculating the radius of a cone?

A: Some common mistakes to avoid when calculating the radius of a cone include:

  • Not using the correct formula for the surface area of a cone
  • Not substituting the relationship between the height and radius correctly
  • Not simplifying the equation correctly
  • Not taking the square root correctly

Conclusion

In this article, we have answered some frequently asked questions related to calculating the radius of a cone. We have provided step-by-step instructions on how to calculate the radius of a cone given its surface area and the relationship between the height and radius. We have also discussed some real-world applications of calculating the radius of a cone and some common mistakes to avoid.

Discussion

Calculating the radius of a cone is an important concept in mathematics and has many real-world applications. It requires a deep understanding of cone geometry and the ability to apply mathematical formulas to solve real-world problems. In this article, we have provided a comprehensive guide to calculating the radius of a cone and have answered some frequently asked questions related to this topic.

References

  • [1] "Geometry of Cones" by Math Open Reference
  • [2] "Surface Area of a Cone" by Math Is Fun
  • [3] "Cones and Spheres" by Khan Academy

Additional Resources

  • [1] "Cones and Cylinders" by IXL
  • [2] "Surface Area and Volume of Cones" by Purplemath
  • [3] "Geometry of Cones" by Wolfram Alpha