A Cylindrical Pipe Is 8 Ft Long And Has A Volume Of $104 \, \text{ft}^3$. Find The Approximate Radius To The Nearest Hundredth Of A Foot.Given:$\[ V = \pi R^2 H \\]where:- \[$ V = 104 \, \text{ft}^3 \$\]- \[$ H = 8 \,

by ADMIN 218 views

=====================================================

Problem Overview

In this problem, we are given a cylindrical pipe with a volume of 104 ft3104 \, \text{ft}^3 and a length of 8 ft. We need to find the approximate radius of the pipe to the nearest hundredth of a foot.

Given Information

  • The volume of the cylindrical pipe is 104 ft3104 \, \text{ft}^3.
  • The length of the cylindrical pipe is 8 ft.
  • The formula for the volume of a cylinder is given by: V=πr2hV = \pi r^2 h.

Formula and Variables

  • VV is the volume of the cylinder.
  • rr is the radius of the cylinder.
  • hh is the height (or length) of the cylinder.
  • π\pi is a mathematical constant approximately equal to 3.14159.

Substituting Given Values

We can substitute the given values into the formula for the volume of a cylinder:

V=πr2h{ V = \pi r^2 h }

Substituting V=104 ft3V = 104 \, \text{ft}^3 and h=8 fth = 8 \, \text{ft}, we get:

104=πr2(8){ 104 = \pi r^2 (8) }

Simplifying the Equation

We can simplify the equation by dividing both sides by 8:

13=πr2{ 13 = \pi r^2 }

Solving for rr

To solve for rr, we can divide both sides by π\pi:

r2=13π{ r^2 = \frac{13}{\pi} }

Taking the square root of both sides, we get:

r=13π{ r = \sqrt{\frac{13}{\pi}} }

Calculating the Value of rr

We can calculate the value of rr using a calculator:

r≈133.14159{ r \approx \sqrt{\frac{13}{3.14159}} }

r≈4.134{ r \approx \sqrt{4.134} }

r≈2.03{ r \approx 2.03 }

Rounding to the Nearest Hundredth

We need to round the value of rr to the nearest hundredth of a foot:

r≈2.03 ft{ r \approx 2.03 \, \text{ft} }

The final answer is 2.03\boxed{2.03}.

Conclusion

In this problem, we used the formula for the volume of a cylinder to find the approximate radius of a cylindrical pipe with a volume of 104 ft3104 \, \text{ft}^3 and a length of 8 ft. We substituted the given values into the formula, simplified the equation, and solved for rr. The final answer is 2.03 ft2.03 \, \text{ft}.

Additional Information

  • The formula for the volume of a cylinder is a fundamental concept in mathematics and is used in a wide range of applications, including engineering, physics, and architecture.
  • The value of π\pi is an irrational number that is approximately equal to 3.14159.
  • The radius of a cylinder is an important parameter that determines the volume and surface area of the cylinder.

References

  • [1] "Mathematics for Engineers and Scientists" by Donald R. Hill
  • [2] "Calculus" by Michael Spivak
  • [3] "Geometry" by I.M. Gelfand

Tags

  • mathematics
  • cylindrical pipe
  • volume
  • radius
  • formula
  • calculation
  • approximation
  • engineering
  • physics
  • architecture

=====================================

Frequently Asked Questions

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

A: The formula for the volume of a cylinder is given by: V=πr2hV = \pi r^2 h.

Q: What is the given information in this problem?

A: The given information is that the volume of the cylindrical pipe is 104 ft3104 \, \text{ft}^3 and the length of the pipe is 8 ft.

Q: How do we substitute the given values into the formula?

A: We substitute the given values into the formula by replacing VV with 104 ft3104 \, \text{ft}^3 and hh with 8 ft.

Q: What is the simplified equation after substituting the given values?

A: The simplified equation is: 13=πr213 = \pi r^2.

Q: How do we solve for rr?

A: We solve for rr by dividing both sides of the equation by π\pi and then taking the square root of both sides.

Q: What is the value of rr?

A: The value of rr is approximately 2.03 ft.

Q: Why do we need to round the value of rr to the nearest hundredth?

A: We need to round the value of rr to the nearest hundredth because the problem asks for the radius to the nearest hundredth of a foot.

Q: What is the final answer?

A: The final answer is 2.03\boxed{2.03}.

Additional Questions and Answers

Q: What is the significance of the value of π\pi in this problem?

A: The value of π\pi is an irrational number that is approximately equal to 3.14159. It is used in the formula for the volume of a cylinder.

Q: How is the formula for the volume of a cylinder used in real-world applications?

A: The formula for the volume of a cylinder is used in a wide range of applications, including engineering, physics, and architecture.

Q: What are some common applications of the formula for the volume of a cylinder?

A: Some common applications of the formula for the volume of a cylinder include calculating the volume of pipes, tanks, and other cylindrical objects.

Q: How can we use the formula for the volume of a cylinder to solve other problems?

A: We can use the formula for the volume of a cylinder to solve other problems by substituting the given values into the formula and then solving for the unknown variable.

Conclusion

In this Q&A article, we have answered some of the most frequently asked questions about the problem of finding the radius of a cylindrical pipe with a volume of 104 ft3104 \, \text{ft}^3 and a length of 8 ft. We have also discussed some additional questions and answers related to the formula for the volume of a cylinder and its applications.

Tags

  • mathematics
  • cylindrical pipe
  • volume
  • radius
  • formula
  • calculation
  • approximation
  • engineering
  • physics
  • architecture

References

  • [1] "Mathematics for Engineers and Scientists" by Donald R. Hill
  • [2] "Calculus" by Michael Spivak
  • [3] "Geometry" by I.M. Gelfand