The Transportation Department Needs To Install A Railing Along The Outer Curve Of A Road That Is 20 Feet Wide. What Is The Length Of The Outer Curve To The Nearest Foot? Use The Value $\pi = 3.14$.A. 68 Feet B. 74 Feet C. 86 Feet D. 92 Feet

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Introduction

The transportation department is tasked with installing a railing along the outer curve of a road that is 20 feet wide. To determine the length of the outer curve, we need to use the formula for the circumference of a circle, which is given by C = 2πr, where C is the circumference and r is the radius of the circle. In this case, the width of the road is given as 20 feet, which is equal to the diameter of the circle. We will use the value of π = 3.14 to calculate the length of the outer curve.

The Formula for Circumference

The formula for the circumference of a circle is given by C = 2Ï€r, where C is the circumference and r is the radius of the circle. Since the width of the road is given as 20 feet, which is equal to the diameter of the circle, we can use the formula to calculate the circumference.

Calculating the Circumference

To calculate the circumference, we need to first find the radius of the circle. Since the diameter is given as 20 feet, we can find the radius by dividing the diameter by 2.

diameter = 20
radius = diameter / 2
print(radius)

The radius of the circle is 10 feet. Now, we can use the formula for the circumference to calculate the length of the outer curve.

import math

pi = 3.14
circumference = 2 * pi * radius
print(circumference)

The length of the outer curve is approximately 62.8 feet. However, we need to round this value to the nearest foot.

Rounding to the Nearest Foot

To round the value to the nearest foot, we can use the round() function in Python.

rounded_circumference = round(circumference)
print(rounded_circumference)

The length of the outer curve to the nearest foot is 63 feet.

Conclusion

In conclusion, the length of the outer curve to the nearest foot is 63 feet. This value can be used by the transportation department to determine the length of the railing that needs to be installed along the outer curve of the road.

Comparison with Answer Choices

Let's compare our answer with the answer choices provided.

  • A. 68 feet
  • B. 74 feet
  • C. 86 feet
  • D. 92 feet

Our answer, 63 feet, is not among the answer choices. However, we can see that the closest answer choice is A. 68 feet.

Discussion

The transportation department needs to install a railing along the outer curve of a road that is 20 feet wide. To determine the length of the outer curve, we used the formula for the circumference of a circle, which is given by C = 2πr. We calculated the circumference using the value of π = 3.14 and found that the length of the outer curve to the nearest foot is 63 feet. This value can be used by the transportation department to determine the length of the railing that needs to be installed along the outer curve of the road.

Mathematical Concepts

This problem involves the following mathematical concepts:

  • Circumference of a circle
  • Formula for circumference: C = 2Ï€r
  • Radius of a circle
  • Diameter of a circle
  • Rounding to the nearest foot

Real-World Applications

This problem has real-world applications in the field of transportation engineering. The transportation department needs to install a railing along the outer curve of a road to ensure the safety of drivers and pedestrians. By calculating the length of the outer curve, the department can determine the length of the railing that needs to be installed.

Future Research Directions

Future research directions in this area could include:

  • Developing more accurate methods for calculating the length of the outer curve
  • Investigating the effects of different values of Ï€ on the calculation
  • Exploring the use of other mathematical formulas for calculating the length of the outer curve

Conclusion

Q&A: Frequently Asked Questions

Q: What is the formula for the circumference of a circle? A: The formula for the circumference of a circle is given by C = 2Ï€r, where C is the circumference and r is the radius of the circle.

Q: How do I calculate the radius of a circle? A: To calculate the radius of a circle, you need to divide the diameter of the circle by 2. In this case, the diameter is given as 20 feet, so the radius is 10 feet.

Q: What is the value of π used in this problem? A: The value of π used in this problem is 3.14.

Q: How do I calculate the length of the outer curve? A: To calculate the length of the outer curve, you need to use the formula for the circumference of a circle, which is given by C = 2Ï€r. You can then round the value to the nearest foot.

Q: What is the length of the outer curve to the nearest foot? A: The length of the outer curve to the nearest foot is 63 feet.

Q: Why is it important to calculate the length of the outer curve? A: It is important to calculate the length of the outer curve because it determines the length of the railing that needs to be installed along the outer curve of the road. This ensures the safety of drivers and pedestrians.

Q: What are some real-world applications of this problem? A: Some real-world applications of this problem include:

  • Transportation engineering: Calculating the length of the outer curve is essential for designing and installing railings along roads and highways.
  • Civil engineering: Calculating the length of the outer curve is also important for designing and building bridges and other infrastructure projects.
  • Architecture: Calculating the length of the outer curve is necessary for designing and building buildings and other structures.

Q: What are some future research directions in this area? A: Some future research directions in this area could include:

  • Developing more accurate methods for calculating the length of the outer curve
  • Investigating the effects of different values of Ï€ on the calculation
  • Exploring the use of other mathematical formulas for calculating the length of the outer curve

Q: What mathematical concepts are involved in this problem? A: The mathematical concepts involved in this problem include:

  • Circumference of a circle
  • Formula for circumference: C = 2Ï€r
  • Radius of a circle
  • Diameter of a circle
  • Rounding to the nearest foot

Conclusion

In conclusion, the length of the outer curve to the nearest foot is 63 feet. This value can be used by the transportation department to determine the length of the railing that needs to be installed along the outer curve of the road. The mathematical concepts involved in this problem include the circumference of a circle, formula for circumference, radius of a circle, diameter of a circle, and rounding to the nearest foot. This problem has real-world applications in the field of transportation engineering and has the potential for future research directions.

Additional Resources

For more information on this topic, please refer to the following resources:

  • [1] "Circumference of a Circle" by Math Open Reference
  • [2] "Formula for Circumference" by Mathway
  • [3] "Radius of a Circle" by Khan Academy

References

[1] Math Open Reference. (n.d.). Circumference of a Circle. Retrieved from https://www.mathopenref.com/circlecircumference.html

[2] Mathway. (n.d.). Formula for Circumference. Retrieved from https://www.mathway.com/answers/geometry/circle/circumference/1

[3] Khan Academy. (n.d.). Radius of a Circle. Retrieved from https://www.khanacademy.org/math/geometry/geometry-circles/v/radius-of-a-circle