A Parabolic Listening Device Is Shaped Such That The Depth Of The Bowl Is Determined By The Curve $D(H) = 0.145r^2$. What Is The Radius When The Depth Is 8 Inches? Round To The Nearest Thousandth.A. 4.417 Inches B. 5.252 Inches C. 7.428

Introduction

A parabolic listening device is a type of acoustic device that uses a parabolic shape to focus sound waves. The depth of the bowl of the device is determined by the curve $D(H) = 0.145r^2$, where $r$ is the radius of the bowl. In this article, we will calculate the radius of the bowl when the depth is 8 inches.

The Equation of the Curve

The equation of the curve is given by $D(H) = 0.145r^2$. This equation represents the relationship between the depth of the bowl and the radius of the bowl.

Solving for the Radius

To find the radius when the depth is 8 inches, we need to substitute $D(H) = 8$ into the equation and solve for $r$.

$$8 = 0.145r^2$$

To solve for $r$, we can divide both sides of the equation by 0.145.

$$\frac{8}{0.145} = r^2$$

$$55.172 = r^2$$

Now, we can take the square root of both sides of the equation to find the value of $r$.

$$r = \sqrt{55.172}$$

$$r \approx 7.428$$

Rounding to the Nearest Thousandth

The value of $r$ is approximately 7.428. However, we need to round this value to the nearest thousandth.

$$r \approx 7.428 \approx 7.428$$

Conclusion

In this article, we calculated the radius of the bowl of a parabolic listening device when the depth is 8 inches. We used the equation $D(H) = 0.145r^2$ to find the value of $r$. The value of $r$ is approximately 7.428, which is the correct answer.

Discussion

The calculation of the radius of the bowl of a parabolic listening device is an important problem in mathematics. The parabolic shape of the device is used to focus sound waves, and the depth of the bowl is determined by the curve $D(H) = 0.145r^2$. In this article, we used this equation to find the value of $r$ when the depth is 8 inches.

Mathematical Concepts

This problem involves the following mathematical concepts:

• Equations: The equation $D(H) = 0.145r^2$ represents the relationship between the depth of the bowl and the radius of the bowl.
• Solving for a variable: We solved for the variable $r$ by substituting $D(H) = 8$ into the equation and solving for $r$.
• Square roots: We took the square root of both sides of the equation to find the value of $r$.
• Rounding: We rounded the value of $r$ to the nearest thousandth.

Real-World Applications

The calculation of the radius of the bowl of a parabolic listening device has real-world applications in the field of acoustics. The parabolic shape of the device is used to focus sound waves, and the depth of the bowl is determined by the curve $D(H) = 0.145r^2$. This equation is used to design and build parabolic listening devices that can focus sound waves with high accuracy.

Future Research Directions

Future research directions in this area could include:

• Designing parabolic listening devices with different shapes: Researchers could design and build parabolic listening devices with different shapes and sizes to see how they affect the focusing of sound waves.
• Investigating the effects of different materials on the focusing of sound waves: Researchers could investigate how different materials affect the focusing of sound waves in parabolic listening devices.
• Developing new equations to describe the relationship between the depth of the bowl and the radius of the bowl: Researchers could develop new equations to describe the relationship between the depth of the bowl and the radius of the bowl, and use these equations to design and build more accurate parabolic listening devices.
  A Parabolic Listening Device: Q&A =====================================

Introduction

In our previous article, we calculated the radius of the bowl of a parabolic listening device when the depth is 8 inches. We used the equation $D(H) = 0.145r^2$ to find the value of $r$. In this article, we will answer some frequently asked questions about parabolic listening devices and the calculation of the radius of the bowl.

Q: What is a parabolic listening device?

A: A parabolic listening device is a type of acoustic device that uses a parabolic shape to focus sound waves. The parabolic shape of the device is used to collect and focus sound waves, allowing for more accurate listening and recording.

Q: How does the parabolic shape of the device affect the focusing of sound waves?

A: The parabolic shape of the device is used to collect and focus sound waves. The shape of the device is designed to reflect sound waves back to a single point, allowing for more accurate listening and recording.

Q: What is the equation $D(H) = 0.145r^2$ used for?

A: The equation $D(H) = 0.145r^2$ is used to describe the relationship between the depth of the bowl and the radius of the bowl of a parabolic listening device. This equation is used to design and build parabolic listening devices that can focus sound waves with high accuracy.

Q: How do you calculate the radius of the bowl of a parabolic listening device?

A: To calculate the radius of the bowl of a parabolic listening device, you need to substitute the depth of the bowl into the equation $D(H) = 0.145r^2$ and solve for $r$. This involves taking the square root of both sides of the equation to find the value of $r$.

Q: What is the significance of the value of $r$ in the calculation of the radius of the bowl?

A: The value of $r$ represents the radius of the bowl of the parabolic listening device. This value is important because it determines the accuracy of the device in focusing sound waves.

Q: Can you provide an example of how to calculate the radius of the bowl of a parabolic listening device?

A: Yes, let's say we want to calculate the radius of the bowl of a parabolic listening device when the depth is 8 inches. We can substitute $D(H) = 8$ into the equation $D(H) = 0.145r^2$ and solve for $r$.

$$8 = 0.145r^2$$

$$\frac{8}{0.145} = r^2$$

$$55.172 = r^2$$

$$r = \sqrt{55.172}$$

$$r \approx 7.428$$

Q: What are some real-world applications of parabolic listening devices?

A: Parabolic listening devices have a number of real-world applications, including:

• Acoustic research: Parabolic listening devices are used in acoustic research to study the properties of sound waves.
• Sound recording: Parabolic listening devices are used in sound recording to capture high-quality audio.
• Listening devices for the deaf: Parabolic listening devices are used to help people with hearing impairments to listen and communicate more effectively.

Q: What are some potential limitations of parabolic listening devices?

A: Some potential limitations of parabolic listening devices include:

• Cost: Parabolic listening devices can be expensive to purchase and maintain.
• Size: Parabolic listening devices can be large and cumbersome to use.
• Interference: Parabolic listening devices can be affected by interference from other sound sources.

Conclusion

In this article, we have answered some frequently asked questions about parabolic listening devices and the calculation of the radius of the bowl. We have discussed the equation $D(H) = 0.145r^2$ and how it is used to describe the relationship between the depth of the bowl and the radius of the bowl. We have also discussed some real-world applications of parabolic listening devices and some potential limitations of these devices.