An Electrician Has To Repair An Electric Fault At A Point 1·5 M Below The Top Of A Pole, Which Is 5 M High. What Should Be The Length Of The Ladder He Has To Use Which When Inclined At An Angle Of 60 To The Horizontal, Would Enable Him To Reach The

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Introduction

As an electrician, one of the most critical tasks is to repair electrical faults in hard-to-reach areas. In this scenario, we have an electrician who needs to repair an electric fault at a point 1.5 m below the top of a pole, which is 5 m high. The electrician must use a ladder that is inclined at an angle of 60° to the horizontal to reach the fault. In this article, we will explore the calculations required to determine the ideal length of the ladder that the electrician needs to use.

Understanding the Problem

To solve this problem, we need to understand the geometry involved. We have a right-angled triangle formed by the pole, the ground, and the ladder. The height of the pole (opposite side) is 5 m, and the distance from the base of the pole to the point where the fault needs to be repaired (adjacent side) is 1.5 m. The angle between the ladder and the horizontal is 60°.

Calculating the Ideal Ladder Length

To calculate the ideal ladder length, we can use the sine function, which is defined as the ratio of the opposite side to the hypotenuse. In this case, the opposite side is the height of the pole (5 m), and the angle is 60°. We can use the sine function to find the length of the hypotenuse (ladder length).

sin(60°) = opposite side / hypotenuse sin(60°) = 5 m / hypotenuse

To find the hypotenuse, we can rearrange the equation:

hypotenuse = opposite side / sin(60°) hypotenuse = 5 m / sin(60°)

Using a calculator, we can find the value of sin(60°), which is approximately 0.866. Now, we can plug in the value:

hypotenuse = 5 m / 0.866 hypotenuse ≈ 5.77 m

However, this is not the ideal ladder length, as it does not take into account the distance from the base of the pole to the point where the fault needs to be repaired (1.5 m). To find the ideal ladder length, we need to use the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

a² + b² = c²

In this case, a is the distance from the base of the pole to the point where the fault needs to be repaired (1.5 m), b is the height of the pole (5 m), and c is the ideal ladder length. We can plug in the values:

(1.5 m)² + (5 m)² = c² 2.25 m² + 25 m² = c² 27.25 m² = c²

Now, we can take the square root of both sides to find the ideal ladder length:

c = √27.25 m² c ≈ 5.23 m

Conclusion

In conclusion, the ideal ladder length that the electrician needs to use is approximately 5.23 m. This length takes into account the distance from the base of the pole to the point where the fault needs to be repaired (1.5 m) and the height of the pole (5 m). By using a ladder of this length, the electrician can reach the fault at an angle of 60° to the horizontal.

Real-World Applications

This problem has real-world applications in various fields, including construction, electrical work, and search and rescue operations. In these fields, it is essential to calculate the ideal ladder length to ensure safe and efficient access to hard-to-reach areas.

Safety Considerations

When working with ladders, it is essential to consider safety factors, such as the weight capacity of the ladder, the surface on which the ladder is placed, and the angle at which the ladder is inclined. In this scenario, the electrician must ensure that the ladder is placed on a stable surface and that the angle is not too steep, which could cause the ladder to tip over.

Future Developments

In the future, it would be beneficial to develop more advanced tools and techniques for calculating ideal ladder lengths. This could include the use of computer-aided design (CAD) software, which can simulate various scenarios and provide more accurate calculations.

References

  • [1] "Ladder Safety" by the Occupational Safety and Health Administration (OSHA)
  • [2] "Calculating Ladder Lengths" by the International Association of Electrical Inspectors (IAEI)
  • [3] "Geometry and Trigonometry" by the American Mathematical Society (AMS)

Glossary

  • Hypotenuse: The longest side of a right-angled triangle, opposite the right angle.
  • Opposite side: The side of a right-angled triangle that is opposite the angle being measured.
  • Adjacent side: The side of a right-angled triangle that is adjacent to the angle being measured.
  • Sine function: A trigonometric function that is defined as the ratio of the opposite side to the hypotenuse.
  • Pythagorean theorem: A mathematical formula that describes the relationship between the lengths of the sides of a right-angled triangle.

Introduction

Calculating ideal ladder lengths is a critical aspect of various industries, including construction, electrical work, and search and rescue operations. In our previous article, we explored the calculations required to determine the ideal ladder length for an electrician who needs to repair an electric fault at a point 1.5 m below the top of a pole, which is 5 m high. In this article, we will address some of the most frequently asked questions related to calculating ideal ladder lengths.

Q&A

Q: What is the ideal ladder length for a 5 m high pole with a 1.5 m distance from the base to the point where the fault needs to be repaired?

A: The ideal ladder length is approximately 5.23 m, as calculated using the Pythagorean theorem.

Q: How do I calculate the ideal ladder length for a pole of a different height?

A: To calculate the ideal ladder length for a pole of a different height, you can use the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Q: What is the significance of the angle at which the ladder is inclined?

A: The angle at which the ladder is inclined is critical in determining the ideal ladder length. A steeper angle requires a longer ladder, while a shallower angle requires a shorter ladder.

Q: How do I ensure that the ladder is placed on a stable surface?

A: To ensure that the ladder is placed on a stable surface, you should:

  • Check the surface for any obstacles or hazards
  • Use ladder levelers or stabilizers to prevent the ladder from tipping over
  • Ensure that the ladder is placed on a firm, level surface

Q: What are some safety considerations when working with ladders?

A: Some safety considerations when working with ladders include:

  • Ensuring that the ladder is rated for the weight of the person using it
  • Avoiding overreaching or leaning over the side of the ladder
  • Keeping the ladder at a 75° angle or less
  • Using ladder safety accessories, such as ladder levelers or stabilizers

Q: Can I use a ladder calculator to determine the ideal ladder length?

A: Yes, you can use a ladder calculator to determine the ideal ladder length. Ladder calculators are available online or as mobile apps and can simplify the calculations required to determine the ideal ladder length.

Q: What are some real-world applications of calculating ideal ladder lengths?

A: Some real-world applications of calculating ideal ladder lengths include:

  • Construction: Calculating ideal ladder lengths is critical in construction to ensure safe and efficient access to hard-to-reach areas.
  • Electrical work: Calculating ideal ladder lengths is essential in electrical work to ensure safe and efficient access to electrical equipment and components.
  • Search and rescue operations: Calculating ideal ladder lengths is critical in search and rescue operations to ensure safe and efficient access to hard-to-reach areas.

Conclusion

Calculating ideal ladder lengths is a critical aspect of various industries, including construction, electrical work, and search and rescue operations. By understanding the calculations required to determine the ideal ladder length, you can ensure safe and efficient access to hard-to-reach areas. Remember to always consider safety factors, such as the weight capacity of the ladder, the surface on which the ladder is placed, and the angle at which the ladder is inclined.

References

  • [1] "Ladder Safety" by the Occupational Safety and Health Administration (OSHA)
  • [2] "Calculating Ladder Lengths" by the International Association of Electrical Inspectors (IAEI)
  • [3] "Geometry and Trigonometry" by the American Mathematical Society (AMS)

Glossary

  • Hypotenuse: The longest side of a right-angled triangle, opposite the right angle.
  • Opposite side: The side of a right-angled triangle that is opposite the angle being measured.
  • Adjacent side: The side of a right-angled triangle that is adjacent to the angle being measured.
  • Sine function: A trigonometric function that is defined as the ratio of the opposite side to the hypotenuse.
  • Pythagorean theorem: A mathematical formula that describes the relationship between the lengths of the sides of a right-angled triangle.