The Diagram Below Shows A Laddder Leaning Against An Electric Pole. Calculate I) X Ii) Y 16m 37.6° Y X​

by ADMIN 104 views

Introduction

In this problem, we are given a diagram of a ladder leaning against an electric pole. The ladder forms a right-angled triangle with the ground and the pole. We are asked to calculate the values of x and y, which represent the distance from the base of the ladder to the point where it touches the ground and the height of the ladder against the pole, respectively. To solve this problem, we will use trigonometric ratios and the properties of right-angled triangles.

Calculating x

To calculate the value of x, we can use the sine function, which relates the angle, the opposite side (y), and the hypotenuse (the length of the ladder). The sine function is defined as:

sin(θ) = opposite side / hypotenuse

In this case, the angle is 37.6°, the opposite side is y, and the hypotenuse is the length of the ladder, which is 16m. We can rearrange the formula to solve for y:

y = sin(37.6°) × 16

Using a calculator, we can find the value of sin(37.6°) and multiply it by 16 to get the value of y:

y ≈ 0.615 × 16 y ≈ 9.84m

Now that we have the value of y, we can use the Pythagorean theorem to find the value of x. The Pythagorean theorem states that:

a² + b² = c²

where a and b are the legs of the right-angled triangle, and c is the hypotenuse. In this case, a is x, b is y, and c is the length of the ladder (16m). We can plug in the values we know and solve for x:

x² + y² = 16² x² + (9.84)² = 16² x² + 97.00 = 256 x² = 159 x ≈ √159 x ≈ 12.61m

Calculating y

To calculate the value of y, we can use the cosine function, which relates the angle, the adjacent side (x), and the hypotenuse (the length of the ladder). The cosine function is defined as:

cos(θ) = adjacent side / hypotenuse

In this case, the angle is 37.6°, the adjacent side is x, and the hypotenuse is the length of the ladder, which is 16m. We can rearrange the formula to solve for x:

x = cos(37.6°) × 16

Using a calculator, we can find the value of cos(37.6°) and multiply it by 16 to get the value of x:

x ≈ 0.795 × 16 x ≈ 12.75m

Now that we have the value of x, we can use the Pythagorean theorem to find the value of y. We can plug in the values we know and solve for y:

x² + y² = 16² (12.75)² + y² = 16² 162.56 + y² = 256 y² = 93.44 y ≈ √93.44 y ≈ 9.68m

Conclusion

In this problem, we used trigonometric ratios and the properties of right-angled triangles to calculate the values of x and y. We found that x ≈ 12.61m and y ≈ 9.84m. These values represent the distance from the base of the ladder to the point where it touches the ground and the height of the ladder against the pole, respectively.

Discussion

This problem is a classic example of a right-angled triangle problem, where we use trigonometric ratios to find the lengths of the sides. The sine and cosine functions are used to relate the angle, the opposite side, and the hypotenuse, and the Pythagorean theorem is used to find the length of the other side. This type of problem is commonly encountered in mathematics and physics, and is an essential tool for solving problems involving right-angled triangles.

Applications

This problem has many real-world applications, such as:

  • Building design: Architects use trigonometry to design buildings and ensure that they are structurally sound.
  • Physics: Physicists use trigonometry to describe the motion of objects and calculate distances and velocities.
  • Engineering: Engineers use trigonometry to design and build bridges, roads, and other infrastructure.
  • Surveying: Surveyors use trigonometry to measure distances and angles between landmarks and calculate the position of objects.

Exercises

  1. A ladder is leaning against a wall at an angle of 30°. If the length of the ladder is 10m, find the distance from the base of the ladder to the point where it touches the ground.
  2. A building is 20m tall and has a roof that is 15m long. If the angle between the roof and the ground is 45°, find the distance from the base of the building to the point where the roof touches the ground.
  3. A surveyor is measuring the distance between two landmarks. If the angle between the landmarks is 60° and the distance between the landmarks is 20m, find the distance from the surveyor to the first landmark.

Solutions

  1. x = sin(30°) × 10 x ≈ 0.5 × 10 x ≈ 5m
  2. x = cos(45°) × 20 x ≈ 0.707 × 20 x ≈ 14.14m
  3. x = sin(60°) × 20 x ≈ 0.866 × 20 x ≈ 17.32m

Q: What is the main concept behind this problem?

A: The main concept behind this problem is the use of trigonometric ratios to calculate the lengths of the sides of a right-angled triangle.

Q: What are the given values in this problem?

A: The given values in this problem are the length of the ladder (16m) and the angle between the ladder and the ground (37.6°).

Q: How do we calculate the value of x?

A: We calculate the value of x by using the sine function, which relates the angle, the opposite side (y), and the hypotenuse (the length of the ladder). We can rearrange the formula to solve for y and then use the Pythagorean theorem to find the value of x.

Q: How do we calculate the value of y?

A: We calculate the value of y by using the cosine function, which relates the angle, the adjacent side (x), and the hypotenuse (the length of the ladder). We can rearrange the formula to solve for x and then use the Pythagorean theorem to find the value of y.

Q: What is the relationship between the sine and cosine functions?

A: The sine and cosine functions are related by the Pythagorean identity, which states that sin²(θ) + cos²(θ) = 1.

Q: How do we use the Pythagorean theorem to find the value of x?

A: We use the Pythagorean theorem to find the value of x by plugging in the values we know and solving for x. The Pythagorean theorem states that a² + b² = c², where a and b are the legs of the right-angled triangle, and c is the hypotenuse.

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

A: This problem has many real-world applications, such as building design, physics, engineering, and surveying.

Q: How do we use trigonometry in building design?

A: Architects use trigonometry to design buildings and ensure that they are structurally sound. They use trigonometric ratios to calculate the lengths of the sides of right-angled triangles and to determine the angles between different parts of the building.

Q: How do we use trigonometry in physics?

A: Physicists use trigonometry to describe the motion of objects and calculate distances and velocities. They use trigonometric ratios to calculate the angles between different parts of a system and to determine the forces acting on an object.

Q: How do we use trigonometry in engineering?

A: Engineers use trigonometry to design and build bridges, roads, and other infrastructure. They use trigonometric ratios to calculate the lengths of the sides of right-angled triangles and to determine the angles between different parts of a structure.

Q: How do we use trigonometry in surveying?

A: Surveyors use trigonometry to measure distances and angles between landmarks and calculate the position of objects. They use trigonometric ratios to calculate the lengths of the sides of right-angled triangles and to determine the angles between different parts of a landscape.

Q: What are some common mistakes to avoid when solving this problem?

A: Some common mistakes to avoid when solving this problem include:

  • Not using the correct trigonometric ratio (sine or cosine)
  • Not plugging in the correct values into the formula
  • Not using the Pythagorean theorem to find the value of x
  • Not checking the units of the answer

Q: How do we check the units of the answer?

A: We check the units of the answer by making sure that the units of the answer match the units of the given values. For example, if the given value is in meters, the answer should also be in meters.

Q: What are some tips for solving this problem?

A: Some tips for solving this problem include:

  • Using the correct trigonometric ratio (sine or cosine)
  • Plugging in the correct values into the formula
  • Using the Pythagorean theorem to find the value of x
  • Checking the units of the answer
  • Using a calculator to check the answer

Q: How do we use a calculator to check the answer?

A: We use a calculator to check the answer by plugging in the values we know and solving for x or y. We can also use a calculator to check the units of the answer and to make sure that the answer is reasonable.