A Firefighter Needs To Reach A Window 25 Meters High Using A Ladder. If The Ladder Needs To Be Placed At An Angle Of 75° With The Ground, How Long Does The Ladder Need To Be?A. 28.00 Meters B. 25.88 Meters C. 27.00 Meters D. 24 Meters

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Introduction

In this article, we will delve into a classic problem in mathematics that involves trigonometry and right-angled triangles. A firefighter needs to reach a window that is 25 meters high using a ladder. However, the ladder needs to be placed at an angle of 75° with the ground. The question is, how long does the ladder need to be? This problem requires us to use trigonometric functions to calculate the length of the ladder.

Understanding the Problem

To solve this problem, we need to understand the concept of right-angled triangles and trigonometric functions. A right-angled triangle is a triangle with one angle that is 90°. In this case, the ladder is the hypotenuse of the right-angled triangle, and the height of the window is one of the legs. The angle between the ladder and the ground is given as 75°.

Using Trigonometry to Solve the Problem

To solve this problem, we can use the sine function, which is defined as the ratio of the length of the opposite side (in this case, the height of the window) to the length of the hypotenuse (the ladder). The sine function is given by the formula:

sin(θ) = opposite side / hypotenuse

In this case, we are given the angle (75°) and the opposite side (25 meters). We need to find the length of the hypotenuse (the ladder). We can rearrange the formula to solve for the hypotenuse:

hypotenuse = opposite side / sin(θ)

Calculating the Length of the Ladder

Now that we have the formula, we can plug in the values to calculate the length of the ladder. We are given the angle (75°) and the opposite side (25 meters). We can use a calculator to find the sine of 75°, which is approximately 0.9659. Now we can plug in the values:

hypotenuse = 25 meters / 0.9659

hypotenuse ≈ 25.88 meters

Conclusion

In conclusion, the length of the ladder needed to reach the window 25 meters high at an angle of 75° with the ground is approximately 25.88 meters. This problem requires us to use trigonometric functions to calculate the length of the ladder. We can use the sine function to find the length of the hypotenuse, which is the ladder in this case.

Answer

The correct answer is B. 25.88 meters.

Discussion

This problem is a classic example of a trigonometry problem that involves right-angled triangles. It requires us to use the sine function to calculate the length of the ladder. The problem is relevant to real-life situations, such as a firefighter needing to reach a window to rescue someone. The solution to this problem can be applied to various fields, such as engineering, physics, and architecture.

Real-World Applications

This problem has real-world applications in various fields, such as:

  • Engineering: In engineering, trigonometry is used to calculate the length of beams, bridges, and other structures.
  • Physics: In physics, trigonometry is used to calculate the trajectory of projectiles, such as balls and rockets.
  • Architecture: In architecture, trigonometry is used to calculate the height of buildings and the length of roofs.

Tips and Tricks

Here are some tips and tricks to help you solve this problem:

  • Use a calculator: Use a calculator to find the sine of the angle (75°) and the length of the ladder.
  • Rearrange the formula: Rearrange the formula to solve for the hypotenuse (the ladder).
  • Plug in the values: Plug in the values of the angle and the opposite side (25 meters) into the formula.

Practice Problems

Here are some practice problems to help you practice solving trigonometry problems:

  • Problem 1: A ladder is placed at an angle of 60° with the ground. If the height of the wall is 15 meters, how long is the ladder?
  • Problem 2: A ball is thrown at an angle of 45° with the ground. If the height of the ball is 20 meters, how far does it travel horizontally?
  • Problem 3: A building is 30 meters high. If the angle of elevation of the sun is 30°, how long is the shadow of the building?

Conclusion

In conclusion, this problem requires us to use trigonometric functions to calculate the length of the ladder. We can use the sine function to find the length of the hypotenuse, which is the ladder in this case. The problem has real-world applications in various fields, such as engineering, physics, and architecture. We can use a calculator to find the sine of the angle and the length of the ladder. We can also rearrange the formula to solve for the hypotenuse and plug in the values of the angle and the opposite side.

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Q&A: A Firefighter's Ladder Problem

Q: What is the problem about?

A: The problem is about a firefighter who needs to reach a window that is 25 meters high using a ladder. However, the ladder needs to be placed at an angle of 75° with the ground. The question is, how long does the ladder need to be?

Q: What is the formula to solve this problem?

A: The formula to solve this problem is:

sin(θ) = opposite side / hypotenuse

We can rearrange this formula to solve for the hypotenuse:

hypotenuse = opposite side / sin(θ)

Q: What is the value of the angle (θ) in this problem?

A: The value of the angle (θ) in this problem is 75°.

Q: What is the value of the opposite side in this problem?

A: The value of the opposite side in this problem is 25 meters.

Q: How do we find the sine of the angle (75°)?

A: We can use a calculator to find the sine of the angle (75°). The sine of 75° is approximately 0.9659.

Q: How do we calculate the length of the ladder?

A: We can plug in the values of the angle and the opposite side into the formula:

hypotenuse = 25 meters / 0.9659

hypotenuse ≈ 25.88 meters

Q: What is the correct answer to this problem?

A: The correct answer to this problem is B. 25.88 meters.

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

A: This problem has real-world applications in various fields, such as:

  • Engineering: In engineering, trigonometry is used to calculate the length of beams, bridges, and other structures.
  • Physics: In physics, trigonometry is used to calculate the trajectory of projectiles, such as balls and rockets.
  • Architecture: In architecture, trigonometry is used to calculate the height of buildings and the length of roofs.

Q: What are some tips and tricks to help solve this problem?

A: Here are some tips and tricks to help you solve this problem:

  • Use a calculator: Use a calculator to find the sine of the angle (75°) and the length of the ladder.
  • Rearrange the formula: Rearrange the formula to solve for the hypotenuse (the ladder).
  • Plug in the values: Plug in the values of the angle and the opposite side (25 meters) into the formula.

Q: What are some practice problems to help practice solving trigonometry problems?

A: Here are some practice problems to help you practice solving trigonometry problems:

  • Problem 1: A ladder is placed at an angle of 60° with the ground. If the height of the wall is 15 meters, how long is the ladder?
  • Problem 2: A ball is thrown at an angle of 45° with the ground. If the height of the ball is 20 meters, how far does it travel horizontally?
  • Problem 3: A building is 30 meters high. If the angle of elevation of the sun is 30°, how long is the shadow of the building?

Conclusion

In conclusion, this problem requires us to use trigonometric functions to calculate the length of the ladder. We can use the sine function to find the length of the hypotenuse, which is the ladder in this case. The problem has real-world applications in various fields, such as engineering, physics, and architecture. We can use a calculator to find the sine of the angle and the length of the ladder. We can also rearrange the formula to solve for the hypotenuse and plug in the values of the angle and the opposite side.