From The Top Of A Tree, A Bird Looks Down On A Field Mouse At An Angle Of Depression Of $50^{\circ}$. If The Field Mouse Is 40 Meters From The Base Of The Tree, Find The Vertical Distance From The Ground To The Bird's Eyes. Round The Answer

by ADMIN 243 views

Introduction

In trigonometry, the angle of depression is a concept that is often used to solve problems involving right triangles. The angle of depression is the angle between the horizontal and the line of sight from the top of an object to a point on the ground. In this problem, we are given that a bird is looking down at a field mouse from the top of a tree at an angle of depression of $50^{\circ}$. We are also given that the field mouse is 40 meters from the base of the tree. Our goal is to find the vertical distance from the ground to the bird's eyes.

Understanding the Problem

To solve this problem, we need to understand the concept of the angle of depression and how it relates to the right triangle formed by the tree, the bird, and the field mouse. The angle of depression is the angle between the horizontal and the line of sight from the top of the tree to the field mouse. This angle is given as $50^{\circ}$. We also know that the field mouse is 40 meters from the base of the tree, which means that the horizontal distance from the base of the tree to the field mouse is 40 meters.

Drawing a Diagram

To better understand the problem, let's draw a diagram of the situation. We can draw a right triangle with the tree as the vertical leg, the bird's line of sight as the hypotenuse, and the field mouse as the horizontal leg. The angle of depression is the angle between the horizontal and the line of sight from the top of the tree to the field mouse.

Using Trigonometry to Solve the Problem

To find the vertical distance from the ground to the bird's eyes, we can use the concept of the tangent function. The tangent function is defined as the ratio of the opposite side to the adjacent side in a right triangle. In this case, the opposite side is the vertical distance from the ground to the bird's eyes, and the adjacent side is the horizontal distance from the base of the tree to the field mouse.

Calculating the Vertical Distance

Using the tangent function, we can calculate the vertical distance from the ground to the bird's eyes as follows:

tan⁡(50∘)=oppositeadjacent=vertical distance40\tan(50^{\circ}) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\text{vertical distance}}{40}

Rearranging the equation to solve for the vertical distance, we get:

vertical distance=40×tan⁡(50∘)\text{vertical distance} = 40 \times \tan(50^{\circ})

Evaluating the Expression

To evaluate the expression, we need to calculate the value of $\tan(50^{\circ})$. Using a calculator, we get:

tan⁡(50∘)≈1.1918\tan(50^{\circ}) \approx 1.1918

Substituting this value into the expression, we get:

vertical distance=40×1.1918≈47.672\text{vertical distance} = 40 \times 1.1918 \approx 47.672

Rounding the Answer

To round the answer, we can round the vertical distance to the nearest whole number. Rounding 47.672 to the nearest whole number, we get:

vertical distance≈48\text{vertical distance} \approx 48

Conclusion

In this problem, we used the concept of the angle of depression and the tangent function to find the vertical distance from the ground to the bird's eyes. We calculated the vertical distance using the tangent function and evaluated the expression to get an approximate value. Finally, we rounded the answer to the nearest whole number to get the final answer.

Final Answer

The final answer is: 48\boxed{48}

Introduction

In our previous article, we explored the concept of the angle of depression and how it can be used to solve problems involving right triangles. We used the tangent function to find the vertical distance from the ground to the bird's eyes. In this article, we will answer some common questions related to the problem and provide additional insights.

Q: What is the angle of depression?

A: The angle of depression is the angle between the horizontal and the line of sight from the top of an object to a point on the ground. In this problem, the angle of depression is $50^{\circ}$.

Q: Why is the angle of depression important?

A: The angle of depression is important because it can be used to solve problems involving right triangles. It is a key concept in trigonometry and is used to find the lengths of sides and angles in right triangles.

Q: How do you calculate the angle of depression?

A: To calculate the angle of depression, you need to know the length of the horizontal side and the length of the vertical side. You can use the tangent function to calculate the angle of depression.

Q: What is the tangent function?

A: The tangent function is a trigonometric function that is defined as the ratio of the opposite side to the adjacent side in a right triangle. It is used to find the lengths of sides and angles in right triangles.

Q: How do you use the tangent function to solve problems involving the angle of depression?

A: To use the tangent function to solve problems involving the angle of depression, you need to know the length of the horizontal side and the length of the vertical side. You can use the tangent function to calculate the angle of depression and then use it to find the lengths of sides and angles in the right triangle.

Q: What are some common mistakes to avoid when solving problems involving the angle of depression?

A: Some common mistakes to avoid when solving problems involving the angle of depression include:

  • Not using the correct formula for the tangent function
  • Not using the correct values for the lengths of the sides
  • Not checking the units of the answer
  • Not rounding the answer to the correct number of decimal places

Q: How do you check your answer when solving problems involving the angle of depression?

A: To check your answer when solving problems involving the angle of depression, you can use the following steps:

  • Check that you have used the correct formula for the tangent function
  • Check that you have used the correct values for the lengths of the sides
  • Check that the units of the answer are correct
  • Check that the answer is rounded to the correct number of decimal places

Q: What are some real-world applications of the angle of depression?

A: Some real-world applications of the angle of depression include:

  • Building design: Architects use the angle of depression to design buildings that are safe and functional.
  • Engineering: Engineers use the angle of depression to design bridges, roads, and other infrastructure.
  • Surveying: Surveyors use the angle of depression to measure the distance and angle between two points.
  • Photography: Photographers use the angle of depression to take pictures of objects from different angles.

Conclusion

In this article, we answered some common questions related to the problem and provided additional insights. We discussed the importance of the angle of depression, how to calculate it, and how to use it to solve problems involving right triangles. We also discussed some common mistakes to avoid and how to check your answer when solving problems involving the angle of depression. Finally, we discussed some real-world applications of the angle of depression.

Final Answer

The final answer is: 48\boxed{48}