An Engineer Wants To Know The Volume Of An Object Composed Of A Hemisphere And A Cone. The Height Of The Cone Is Equal To The Diameter Of The Hemisphere, And The Volume Of The Cone Is $12x^3$. What Is The Volume Of The Whole Object?A.

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Introduction

In various engineering applications, it is essential to calculate the volume of complex objects composed of different shapes. In this article, we will focus on finding the volume of an object consisting of a hemisphere and a cone. The height of the cone is equal to the diameter of the hemisphere, and the volume of the cone is given as $12x^3$. Our objective is to determine the total volume of the composite object.

Understanding the Shapes

Before we proceed with the calculation, let's briefly discuss the shapes involved. A hemisphere is half of a sphere, and its volume can be calculated using the formula $\frac{2}{3}\pi r^3$, where $r$ is the radius of the hemisphere. A cone is a three-dimensional shape with a circular base and a pointed top. The volume of a cone can be calculated using the formula $\frac{1}{3}\pi r^2 h$, where $r$ is the radius of the base and $h$ is the height of the cone.

Given Information

We are given that the height of the cone is equal to the diameter of the hemisphere. Let's denote the radius of the hemisphere as $r$. Since the height of the cone is equal to the diameter of the hemisphere, the height of the cone is $2r$. We are also given that the volume of the cone is $12x^3$.

Calculating the Volume of the Hemisphere

To calculate the volume of the hemisphere, we can use the formula $\frac{2}{3}\pi r^3$. However, we need to express the volume of the hemisphere in terms of $x$. Since the height of the cone is equal to the diameter of the hemisphere, we can write $2r = x$. Solving for $r$, we get $r = \frac{x}{2}$. Substituting this value of $r$ into the formula for the volume of the hemisphere, we get:

Vhemisphere=23Ï€(x2)3=16Ï€x3V_{\text{hemisphere}} = \frac{2}{3}\pi \left(\frac{x}{2}\right)^3 = \frac{1}{6}\pi x^3

Calculating the Volume of the Cone

We are given that the volume of the cone is $12x^3$. However, we need to express the volume of the cone in terms of $r$. Since the height of the cone is equal to the diameter of the hemisphere, we can write $2r = x$. Solving for $r$, we get $r = \frac{x}{2}$. Substituting this value of $r$ into the formula for the volume of the cone, we get:

Vcone=13Ï€(x2)2(2r)=13Ï€(x2)2(x)=112Ï€x3V_{\text{cone}} = \frac{1}{3}\pi \left(\frac{x}{2}\right)^2 \left(2r\right) = \frac{1}{3}\pi \left(\frac{x}{2}\right)^2 \left(x\right) = \frac{1}{12}\pi x^3

However, we are given that the volume of the cone is $12x^3$. This means that our previous calculation is incorrect. Let's re-examine the formula for the volume of the cone:

Vcone=13Ï€r2hV_{\text{cone}} = \frac{1}{3}\pi r^2 h

Substituting $h = 2r$, we get:

Vcone=13Ï€r2(2r)=23Ï€r3V_{\text{cone}} = \frac{1}{3}\pi r^2 \left(2r\right) = \frac{2}{3}\pi r^3

Since the volume of the cone is $12x^3$, we can set up the equation:

23Ï€r3=12x3\frac{2}{3}\pi r^3 = 12x^3

Solving for $r$, we get:

r=18x3Ï€r = \sqrt{\frac{18x^3}{\pi}}

Now that we have the value of $r$, we can calculate the volume of the hemisphere:

Vhemisphere=23Ï€r3=23Ï€(18x3Ï€)3=36x3V_{\text{hemisphere}} = \frac{2}{3}\pi r^3 = \frac{2}{3}\pi \left(\sqrt{\frac{18x^3}{\pi}}\right)^3 = 36x^3

Calculating the Total Volume

The total volume of the composite object is the sum of the volumes of the hemisphere and the cone:

Vtotal=Vhemisphere+Vcone=36x3+12x3=48x3V_{\text{total}} = V_{\text{hemisphere}} + V_{\text{cone}} = 36x^3 + 12x^3 = 48x^3

Therefore, the total volume of the composite object is $48x^3$.

Conclusion

In this article, we calculated the volume of a composite object consisting of a hemisphere and a cone. The height of the cone is equal to the diameter of the hemisphere, and the volume of the cone is given as $12x^3$. We used the formulas for the volumes of a hemisphere and a cone to calculate the total volume of the composite object. The final answer is $48x^3$.

References

  • [1] "Volume of a Hemisphere" by Math Open Reference
  • [2] "Volume of a Cone" by Math Open Reference

Additional Resources

  • [1] "Hemisphere" by Wikipedia
  • [2] "Cone" by Wikipedia
    Frequently Asked Questions (FAQs) =====================================

Q: What is the formula for the volume of a hemisphere?

A: The formula for the volume of a hemisphere is $\frac{2}{3}\pi r^3$, where $r$ is the radius of the hemisphere.

Q: What is the formula for the volume of a cone?

A: The formula for the volume of a cone is $\frac{1}{3}\pi r^2 h$, where $r$ is the radius of the base and $h$ is the height of the cone.

Q: How do I calculate the volume of a composite object consisting of a hemisphere and a cone?

A: To calculate the volume of a composite object consisting of a hemisphere and a cone, you need to calculate the volumes of the hemisphere and the cone separately and then add them together.

Q: What is the relationship between the height of the cone and the diameter of the hemisphere?

A: The height of the cone is equal to the diameter of the hemisphere.

Q: How do I express the volume of the hemisphere in terms of $x$?

A: To express the volume of the hemisphere in terms of $x$, you need to substitute $r = \frac{x}{2}$ into the formula for the volume of the hemisphere.

Q: How do I calculate the total volume of the composite object?

A: To calculate the total volume of the composite object, you need to add the volumes of the hemisphere and the cone together.

Q: What is the final answer for the total volume of the composite object?

A: The final answer for the total volume of the composite object is $48x^3$.

Q: What are some real-world applications of calculating the volume of a composite object?

A: Some real-world applications of calculating the volume of a composite object include:

  • Calculating the volume of a container with a hemispherical bottom and a conical top
  • Calculating the volume of a fuel tank with a hemispherical bottom and a conical top
  • Calculating the volume of a storage container with a hemispherical bottom and a conical top

Q: How do I use the formulas for the volumes of a hemisphere and a cone in real-world applications?

A: To use the formulas for the volumes of a hemisphere and a cone in real-world applications, you need to substitute the given values into the formulas and perform the necessary calculations.

Q: What are some common mistakes to avoid when calculating the volume of a composite object?

A: Some common mistakes to avoid when calculating the volume of a composite object include:

  • Failing to substitute the correct values into the formulas
  • Failing to perform the necessary calculations
  • Failing to check the units of the answer

Q: How do I check my answer for the total volume of the composite object?

A: To check your answer for the total volume of the composite object, you need to:

  • Substitute the given values into the formulas
  • Perform the necessary calculations
  • Check the units of the answer

Q: What are some additional resources for learning more about calculating the volume of a composite object?

A: Some additional resources for learning more about calculating the volume of a composite object include:

  • Online tutorials and videos
  • Textbooks and reference books
  • Online forums and discussion groups

Conclusion

In this article, we have answered some frequently asked questions about calculating the volume of a composite object consisting of a hemisphere and a cone. We have provided formulas and examples to help you understand the concepts and apply them in real-world applications. We hope this article has been helpful in your learning journey.